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Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false?
Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark the Riemann Hypothesis that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age.
It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution.
Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof and those who have been consumed by it.
446 pages, Hardcover
First published January 1, 2003
333 people are currently reading
7813 people want to read
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June 24, 2015
The best popular mathematics book I can recall reading. I had heard about the Riemann Hypothesis a zillion times and never understood what the fuss was about. After going through this book, it all made sense! Requires college-level math, but if you have that, can't recommend too highly.
______________________________
An anecdote from Lambert's biography of Georges Lemaître which may amuse mathematicians. At one stage, young Lemaître was being supervised by the famous number theorist de Vallée-Poussin, who appears in this book. Impressed by his precocious talent, de Vallée-Poussin suggested to Lemaître that he should try to solve the Riemann conjecture as his thesis project.
After about a year of work, Lemaître was forced to admit that he had got nowhere and was allowed to switch to another topic. He eventually justified his supervisor's faith in him by inventing the Big Bang theory and finding the first experimental evidence for it. He also made contributions to spinor algebra which were praised by Einstein. But the Riemann conjecture was just too hard.
_______________________________
Inspired by yesterday's discussion with Nandakishore and Matt and the famous XKCD cartoon (soon to become a book), I wrote the following:
_______________________________
Following on from that, is it possible to explain the Riemann Hypothesis to an eight-year-old? I have discussed this question with Dodecahedron, who very sensibly said after a while that there was no point in theorizing, you just have to try it. I have formulated a plan of attack, which goes through several increasingly challenging steps. Last night, after getting his parents' permission, I tried out Step 1 on Jenkyn, a very smart eight-year-old we know. The conversation went roughly as follows:
- So Jenkyn, do you know what a prime number is?
- Uh-uh.
- Well, think about numbers. You can turn some numbers into rectangles. Like, 8 is 4 by 2. [I illustrated using pieces on a chessboard.] Or 10 is 5 by 2. Or 9 is 3 by 3. You see, some numbers are rectangles.
- Or squares.
- Okay, some of them are squares. But some other numbers don't make rectangles. You see, you can't turn 5 into a rectangle. Or 7.
- Uh-huh.
- Well, those are called prime numbers. So, 5 is prime and 7 is prime. What else do you think would be a prime number? Is 9 prime?
- Uh-uh.
- Is 11 prime?
- Uh-huh.
- So what else is prime?
- 113.
[Pause while all the adults sit and work it out]
- Uh, yes, 113 is prime. How did you know?!
- I dunno. I just did.
______________________________
An anecdote from Lambert's biography of Georges Lemaître which may amuse mathematicians. At one stage, young Lemaître was being supervised by the famous number theorist de Vallée-Poussin, who appears in this book. Impressed by his precocious talent, de Vallée-Poussin suggested to Lemaître that he should try to solve the Riemann conjecture as his thesis project.
After about a year of work, Lemaître was forced to admit that he had got nowhere and was allowed to switch to another topic. He eventually justified his supervisor's faith in him by inventing the Big Bang theory and finding the first experimental evidence for it. He also made contributions to spinor algebra which were praised by Einstein. But the Riemann conjecture was just too hard.
_______________________________
Inspired by yesterday's discussion with Nandakishore and Matt and the famous XKCD cartoon (soon to become a book), I wrote the following:
Can I explain Mr Riemann's very important number idea with only the ten hundred most used words? It is about how many numbers that don't have pieces there are that are less than some number. Mr Riemann found a way to work it out. You add four things together. One of them means you must add a lot of other things first. But where are those things? Mr Riemann thought he knew but he wasn't sure. He said they are all made of things you find on a line and the real part of that line is a half. People have looked for those things. They have found more than a hundred hundred hundred hundred hundred hundred of them. They are all on Mr Riemann's line. But maybe some of the things are not on the line. No one knows. They want to find out and they have looked for more than one hundred years but it is very hard.Maybe someone can improve on this?
_______________________________
Following on from that, is it possible to explain the Riemann Hypothesis to an eight-year-old? I have discussed this question with Dodecahedron, who very sensibly said after a while that there was no point in theorizing, you just have to try it. I have formulated a plan of attack, which goes through several increasingly challenging steps. Last night, after getting his parents' permission, I tried out Step 1 on Jenkyn, a very smart eight-year-old we know. The conversation went roughly as follows:
- So Jenkyn, do you know what a prime number is?
- Uh-uh.
- Well, think about numbers. You can turn some numbers into rectangles. Like, 8 is 4 by 2. [I illustrated using pieces on a chessboard.] Or 10 is 5 by 2. Or 9 is 3 by 3. You see, some numbers are rectangles.
- Or squares.
- Okay, some of them are squares. But some other numbers don't make rectangles. You see, you can't turn 5 into a rectangle. Or 7.
- Uh-huh.
- Well, those are called prime numbers. So, 5 is prime and 7 is prime. What else do you think would be a prime number? Is 9 prime?
- Uh-uh.
- Is 11 prime?
- Uh-huh.
- So what else is prime?
- 113.
[Pause while all the adults sit and work it out]
- Uh, yes, 113 is prime. How did you know?!
- I dunno. I just did.
August 19, 2020
Gift or Neurosis?
Obsession is something with irresistible psychic power. I take Derbyshire’s use of the term literally. Explaining this power is like explaining any other mental abnormality. From the outside, obsession appears irrational. From the inside its logic is compelling and justifies itself entirely. But only because some unstated fundamental premise is where the logic starts. This is the only really interesting thing about obsession; everything else provokes only a sort of mild voyeurism. Finding that premise is what much of classic biography, as well as psychoanalysis, is about. Without the premise, the obsession is merely strange.
I can understand fascination with the topic of prime numbers and the mathematicians who devoted their lives to their unusual properties. I am fascinated by their fascination, as I am by the experiences of the great mystics. With fascination, one can take it or leave it, however; obsession drives one’s being. And as far as I can tell the great mathematicians are remarkably similar in their obsessions to the great mystics. But we know considerably more about what goes on in the minds of the latter, I suppose because their fixation is more common in the general population.
With mathematicians as with mystics, what interests them most, and what they report about to the rest of us, is technique, that is how they get from one state of knowledge or awareness to another. How could it be otherwise? Without technique they are left with nothing really to communicate except subjective reports of their own psychic states. But as we know all too well, the subjective reports of any of us are probably the least reliable source of information about personal development, spiritual or scientific (See postscript below). So they, and we, construct ‘methods� to hide our lack of understanding of our real motivations. The main function of these narratives is not their general applicability, but their role as evidence that we are in fact sane.
The foundational premise behind the logic of the mystic and the mathematician is obscured by the description of their technique, their methods, as it is by their bare biographical details. We learn nothing about the reason for their obsession, nor what justifies it in their own minds. We learn nothing about the event or sequence of events that led to their discoveries. They therefore appear as prodigies, freaks, automatons, professional celebrities or idols to be worshipped rather than human beings intent on achieving some intellectual or spiritual objective. Their talent is apparent; what they’re trying to do with it is not.
Derbyshire is quite candid about the almost complete lack of documentary information regarding the intimate life of Riemann. Aside from the remarkable fact that he began life as essentially a rural hick and ended as one of the most productive mathematician of the 19th century, there is little that can be said about him which is personal. His travels, his appointments, his professional habits explain nothing about the man’s mind and his intentions. To say that his purpose in life was to solve mathematical problems is simply uninteresting, and probably false.
So Derbyshire is forced to create a sort of Whig history of the man by, on the one hand giving the history of relevant mathematical thought and Riemann’s place in it; and on the other hand telling us about other mathematicians and their equally odd obsessional idiosyncrasies. Everything inevitably leads precisely where it had to. The stars converge, and out pops Riemann and his Hypothesis, which has puzzled every generation of mathematicians since, just as Ignatius Loyola’s Spiritual Exercises have puzzled generations of aspiring religious novices.
The combination of mathematical derivations and historical vignettes in Prime Obsession is frustrating because neither ‘track� explains Riemann’s obsession. So it is also with others who constitute Riemann’s oeuvre. A man like Gauss, for example, had no ambition for influence or power. Born an uneducated peasant, he was likeable but taciturn to the point of rudeness. Are we to think that he was simply taken over by his talent as a mystic is taken over by faith? Was there some fundamental youthful insight, perhaps, that guided his life and found its place in academic mathematics? Could these people have decided to be great novelists instead?
I can understand Derbyshire’s excitement when he writes, “The Riemann Hypothesis was born out of an encounter, ...a great fusion, between counting logic and measuring logic. To put it in precise mathematical terms; it arose when some ideas from arithmetic were combined with some from analysis to form a new thing, a new branch of the mathematical tree, analytic number theory.� I get it; it is an important historical moment. But this is how it is interpreted after the fact, not as Riemann viewed the situation. What was Reimann proving about himself, not just with his Hypothesis but with all his work?
It is of course unfair to burden Derbyshire with these sorts of issues. He has his own obsessions. But I am left wanting something which Derbyshire doesn’t provide, perhaps no one can, namely, a reason for why Riemann did what he did, not merely a description of the deeds. Riemann himself noted that there are an infinite number of mathematical hypotheses which are difficult to prove or disprove. Why make this one? And why has it become central? Is there an unstated aesthetic among mathematicians which is provoked by it? Or does it now simply represent a possibility for professional recognition?
Riemann’s obsession, and Derbyshire’s, sit there, therefore, as the behemoth in the room. As Derbyshire says clearly, “mathematical thinking is a deeply unnatural way of thinking, and this is probably why it repels so many people.� This makes someone like Riemann even more interesting. If he is so unnatural, perhaps that’s part of the explanation for the obsession. My suggestion is that this is what needs to be explored, perhaps even more urgently than the Riemann Hypothesis itself.
Postscript: I was reminded while reading this of something written by an academic acquaintance almost 50 years ago. Ian Mitroff in his book The Subjective Side of Science studied the moon rock scientists while they studied the moon rocks. What he found, as I recall, was an ability to predict their conclusions based on their personalities. At this point I have no idea about the validity or follow-on from his work. Perhaps, however, it is he who is responsible for creating my expectations of Derbyshire! See: /book/show/1...
Obsession is something with irresistible psychic power. I take Derbyshire’s use of the term literally. Explaining this power is like explaining any other mental abnormality. From the outside, obsession appears irrational. From the inside its logic is compelling and justifies itself entirely. But only because some unstated fundamental premise is where the logic starts. This is the only really interesting thing about obsession; everything else provokes only a sort of mild voyeurism. Finding that premise is what much of classic biography, as well as psychoanalysis, is about. Without the premise, the obsession is merely strange.
I can understand fascination with the topic of prime numbers and the mathematicians who devoted their lives to their unusual properties. I am fascinated by their fascination, as I am by the experiences of the great mystics. With fascination, one can take it or leave it, however; obsession drives one’s being. And as far as I can tell the great mathematicians are remarkably similar in their obsessions to the great mystics. But we know considerably more about what goes on in the minds of the latter, I suppose because their fixation is more common in the general population.
With mathematicians as with mystics, what interests them most, and what they report about to the rest of us, is technique, that is how they get from one state of knowledge or awareness to another. How could it be otherwise? Without technique they are left with nothing really to communicate except subjective reports of their own psychic states. But as we know all too well, the subjective reports of any of us are probably the least reliable source of information about personal development, spiritual or scientific (See postscript below). So they, and we, construct ‘methods� to hide our lack of understanding of our real motivations. The main function of these narratives is not their general applicability, but their role as evidence that we are in fact sane.
The foundational premise behind the logic of the mystic and the mathematician is obscured by the description of their technique, their methods, as it is by their bare biographical details. We learn nothing about the reason for their obsession, nor what justifies it in their own minds. We learn nothing about the event or sequence of events that led to their discoveries. They therefore appear as prodigies, freaks, automatons, professional celebrities or idols to be worshipped rather than human beings intent on achieving some intellectual or spiritual objective. Their talent is apparent; what they’re trying to do with it is not.
Derbyshire is quite candid about the almost complete lack of documentary information regarding the intimate life of Riemann. Aside from the remarkable fact that he began life as essentially a rural hick and ended as one of the most productive mathematician of the 19th century, there is little that can be said about him which is personal. His travels, his appointments, his professional habits explain nothing about the man’s mind and his intentions. To say that his purpose in life was to solve mathematical problems is simply uninteresting, and probably false.
So Derbyshire is forced to create a sort of Whig history of the man by, on the one hand giving the history of relevant mathematical thought and Riemann’s place in it; and on the other hand telling us about other mathematicians and their equally odd obsessional idiosyncrasies. Everything inevitably leads precisely where it had to. The stars converge, and out pops Riemann and his Hypothesis, which has puzzled every generation of mathematicians since, just as Ignatius Loyola’s Spiritual Exercises have puzzled generations of aspiring religious novices.
The combination of mathematical derivations and historical vignettes in Prime Obsession is frustrating because neither ‘track� explains Riemann’s obsession. So it is also with others who constitute Riemann’s oeuvre. A man like Gauss, for example, had no ambition for influence or power. Born an uneducated peasant, he was likeable but taciturn to the point of rudeness. Are we to think that he was simply taken over by his talent as a mystic is taken over by faith? Was there some fundamental youthful insight, perhaps, that guided his life and found its place in academic mathematics? Could these people have decided to be great novelists instead?
I can understand Derbyshire’s excitement when he writes, “The Riemann Hypothesis was born out of an encounter, ...a great fusion, between counting logic and measuring logic. To put it in precise mathematical terms; it arose when some ideas from arithmetic were combined with some from analysis to form a new thing, a new branch of the mathematical tree, analytic number theory.� I get it; it is an important historical moment. But this is how it is interpreted after the fact, not as Riemann viewed the situation. What was Reimann proving about himself, not just with his Hypothesis but with all his work?
It is of course unfair to burden Derbyshire with these sorts of issues. He has his own obsessions. But I am left wanting something which Derbyshire doesn’t provide, perhaps no one can, namely, a reason for why Riemann did what he did, not merely a description of the deeds. Riemann himself noted that there are an infinite number of mathematical hypotheses which are difficult to prove or disprove. Why make this one? And why has it become central? Is there an unstated aesthetic among mathematicians which is provoked by it? Or does it now simply represent a possibility for professional recognition?
Riemann’s obsession, and Derbyshire’s, sit there, therefore, as the behemoth in the room. As Derbyshire says clearly, “mathematical thinking is a deeply unnatural way of thinking, and this is probably why it repels so many people.� This makes someone like Riemann even more interesting. If he is so unnatural, perhaps that’s part of the explanation for the obsession. My suggestion is that this is what needs to be explored, perhaps even more urgently than the Riemann Hypothesis itself.
Postscript: I was reminded while reading this of something written by an academic acquaintance almost 50 years ago. Ian Mitroff in his book The Subjective Side of Science studied the moon rock scientists while they studied the moon rocks. What he found, as I recall, was an ability to predict their conclusions based on their personalities. At this point I have no idea about the validity or follow-on from his work. Perhaps, however, it is he who is responsible for creating my expectations of Derbyshire! See: /book/show/1...
October 15, 2009
You remember the smartest kid in your high school calculus class? Remember the math major in your college dorm, the one doing advanced physics with more Greek symbols than Roman numerals? Both brainiacs at the time, right? Well, the book Prime Obsession deals with mathematical concepts magnitudes of order more complex than those brainiacs could ever wish to comprehend. John Derbyshire describes the Riemann Hypothesis (RH) and the mathematical titans that have tried unsuccessfully to prove the hypothesis since 1859. Open the book and you'll be transported to the rarefied air of the Prime Number Theorem and the notorious zeta function.
Derbyshire does a nice shuffle between chapters, alternating between math concepts and biographical background. This does two things: it guides you slowly through the math of Greek symbols and presents the human side of the computational oracle known as Georg Friedrich Bernhard Riemann. I'm not a math guy. I don't even enjoy math. However, after struggling in several higher level calculus classes in college, I have a humble appreciation for and stark awe of folks who can turn, transform, and invert insanely dense and convoluted proofs and theorems. I also realize math is hidden behind many of the technological advances we take for granted as our lives get easier. I don't stop to make the connections between prime number theory and secure internet transactions, but I know it's there, below, hidden, operating, critical to my life.
When I saw several books about the RH, I read a dustcover and, at once, needed to be educated about a theorem that's remained relatively unchanged, and is still only as advanced as when it was presented by Riemann 150 years ago, despite Cray-1 supercomputers crunching possible solutions for 40 years and programmed by myriad math geniuses produced by the 20th century. If it hasn't been solved, then I thought—having never heard of the RH before—Riemann was either a genius among geniuses, or he stumbled upon something mankind was perhaps not to discover for many, many generations to come. Either way, I figured the book would give me a few buzz words to finesse around a cocktail party in the extraordinary event math was discussed over finger foods and wine.
I picked up some buzz words, indeed, and I can use them to parry myself through the initial awkward questions about math. However, I'd have to duck and run from the party because Derbyshire didn't do the best job making the RH understandable. The author is a mathematician, and though he promised to use nothing more advanced than calculus to explain the RH, he was quickly over my head. I know general calculus (quadratic polynomials, convergent & divergent limit functions, matricies, harmonic series, derivation & integration). But, let me walk you through what Derbyshire introduced to me in some 350-odd pages.
To give Derbyshire credit, what Prime Obsession attempts to do is chronologize the different attempts by which mathematicians have tried to solve the RH. Our 20th century boys have not been completely stumped. They've made minor inroads, cracking stone off the monolithic RH, using tactics, techniques, and procedures that are au courant at the time. The book shows calculus, geometry, analytical theory, physics, quantam theory—all specific modus operendi that offer tools of analysis, and which could each stand to benefit from a solution to the RH.
Still, this book is not for the math novice. Also, I think Derbyshire fell short presenting, in plain language, what exactly the RH would prove, and if solved, how it would change the world of mathematics. 3 stars for a solid book, but it won't appear on anybody's favorite list.
New word: ruction
Derbyshire does a nice shuffle between chapters, alternating between math concepts and biographical background. This does two things: it guides you slowly through the math of Greek symbols and presents the human side of the computational oracle known as Georg Friedrich Bernhard Riemann. I'm not a math guy. I don't even enjoy math. However, after struggling in several higher level calculus classes in college, I have a humble appreciation for and stark awe of folks who can turn, transform, and invert insanely dense and convoluted proofs and theorems. I also realize math is hidden behind many of the technological advances we take for granted as our lives get easier. I don't stop to make the connections between prime number theory and secure internet transactions, but I know it's there, below, hidden, operating, critical to my life.
When I saw several books about the RH, I read a dustcover and, at once, needed to be educated about a theorem that's remained relatively unchanged, and is still only as advanced as when it was presented by Riemann 150 years ago, despite Cray-1 supercomputers crunching possible solutions for 40 years and programmed by myriad math geniuses produced by the 20th century. If it hasn't been solved, then I thought—having never heard of the RH before—Riemann was either a genius among geniuses, or he stumbled upon something mankind was perhaps not to discover for many, many generations to come. Either way, I figured the book would give me a few buzz words to finesse around a cocktail party in the extraordinary event math was discussed over finger foods and wine.
I picked up some buzz words, indeed, and I can use them to parry myself through the initial awkward questions about math. However, I'd have to duck and run from the party because Derbyshire didn't do the best job making the RH understandable. The author is a mathematician, and though he promised to use nothing more advanced than calculus to explain the RH, he was quickly over my head. I know general calculus (quadratic polynomials, convergent & divergent limit functions, matricies, harmonic series, derivation & integration). But, let me walk you through what Derbyshire introduced to me in some 350-odd pages.
--pi(N), but not the pi that is 3.14159....
--the Euler-Mascheroni number
--the Rieman Hypothesis: All non-trivial zeros of the zeta funtion have real part one-half
--the Golden Key
--the sieve of Eratosthenes
--the Prime Number Theorem, pi(N)~Li(N)
--complex conjugates
--zeta function critical strip
--Gram's zeros
--Riemann surface
--value plane from the critical line
--Big Oh and Mobius Mu
--Matricies (eigenvalues, trace, characteristic polynomials of)
--operators
--Guassian-random Hermitan matricies
--Guassian Unitary Ensemble
--Chaos theory
--And many, many, many complex, irrational formulas without Roman numerals
To give Derbyshire credit, what Prime Obsession attempts to do is chronologize the different attempts by which mathematicians have tried to solve the RH. Our 20th century boys have not been completely stumped. They've made minor inroads, cracking stone off the monolithic RH, using tactics, techniques, and procedures that are au courant at the time. The book shows calculus, geometry, analytical theory, physics, quantam theory—all specific modus operendi that offer tools of analysis, and which could each stand to benefit from a solution to the RH.
Still, this book is not for the math novice. Also, I think Derbyshire fell short presenting, in plain language, what exactly the RH would prove, and if solved, how it would change the world of mathematics. 3 stars for a solid book, but it won't appear on anybody's favorite list.
New word: ruction
June 13, 2014
First of all, this is pretty well-researched, being a 150 year-long history and all. Also, while quite thorough about the math, it wasn't really that involved. Of course, like all attempts to give a popular account of complicated math, it tread too heavily while not penetrating deep enough. So I was a bit disappointed.
I gave it five stars because despite this, it was a real page turner. The prose is light and clear, and the pace is good. Reminds me of James Gleick's Chaos and Genius, at least in terms of how Derbyshire managed to continually pull off subtle extended metaphors while painting lively portraits of his characters.
So what's the Riemann hypothesis? It turns out to really be quite a trivial-sounding conjecture that Riemann didn't even feel the need to prove in order to come up with some interesting results. Basically, in his analytically continued zeta-function, taking in complex values, he says that all the non trivial zeros have a real part one-half. It's difficult to comprehend why this is significant, and what the significance of those values really are.
More interesting results for me was the expression of zeta in terms of a product solely involving an expression of primes,
and the analytic continuation of the zeta function, which you can generate by taking the more-likely-to-converge eta function and manipulating a bit,
or taking this, which I have no idea how to derive from anything:
Also the gamma function is mind-blowing....
So apart from the math that I still don't really understand, I got a pretty good survey of the interesting historical figures that have populated math since Euler, including Gauss, Dirichlet, Riemann himself, Dedekind, Hilbert, (these five excluding Euler surrounded by the quirky atmosphere of Gottingen which I imagined as a black gothic fort), i forgot what Poussin Hadamard Chebyshev and Landau did, and then of course you have math's Tom and Jerry - Hardy and littlewood, who proved that there were a infinite number of non-trivial zeroes and that Li(x) intersects with pi(n) an infinite number of times (which was very surprising), respectively.
______________Rant______________
1. A big deal was made about big oh of the error term, ie the error term would not exceed modulus x^1/2 logx but okay... I thought Riemann already made a precise formulation of pi(x), given the truth of the hypotheses so... isn't this just a far weaker statement... I did not understand this section at all including the Mobius Mu stuff. So M(k) = O(k^1/2) ????? the inverse of zeta s is equivalent to sqrt(k)??? what does that even mean....??? :(
(Apparently this is a way of saying that if the zeta zeroes don't misbehave, the error term does not misbehave, which was not explicit in Riemann's formulations)
2. So at least we know that the primes are distributed as randomly as one would expect of a Normal or Gaussian distribution and that this may somehow be related to GUEs in QM, which are Hermitian matrices with coeffecients plucked randomly from a normal distribution. But what's the relation??? Idk :(
3. Then there's J(x), which assigns a weighted value to pi of the roots of x. ie
It wasn't explained how or why these values were weighted.
So you can forumlate zeta in terms of this weird shit in terms of J(x)
Idk why it's so important that you'd wanna see zeta in terms of this sort of integral but okay...
4. And fina-fuckin-lly we return to Riemann who manages to formulate J(x) in terms of the zeta function (I thought we already did that????)(and how exactly do you do that?????
So now we have a way of formulating pi(x) in terms of J(x) in terms of zeta (by means of the non-trivial zeta zeroes, in terms of sum of x^all the roots). So we have this really mysterious relation between the values of the zeroes and the more significant "error term" of J(x). Li(x^row) gives you a spiral that winds around pi(the real transcendental number)i and -pi(i)(why pi?) and SOMEHOW this ALL MANAGES TO WORK OUT?
______________________________________
And thus we get a super complicated way of counting the primes (and I sock the guy who says it's elegant, cos wtf is this beast of a thing?).
The zeta zeroes seem just as complicated as the distribution of primes themselves (could one say equivalently complicated), so one would wonder why you would represent one intractable problem in terms of another.
Maybe they're just far too random and unlike most things in the universe of mathematics, don't bear the special transcendental hand print of god. Nevertheless, we choose how to bide our time, and maybe playing games with numbers is just the same as anything else...
I gave it five stars because despite this, it was a real page turner. The prose is light and clear, and the pace is good. Reminds me of James Gleick's Chaos and Genius, at least in terms of how Derbyshire managed to continually pull off subtle extended metaphors while painting lively portraits of his characters.
So what's the Riemann hypothesis? It turns out to really be quite a trivial-sounding conjecture that Riemann didn't even feel the need to prove in order to come up with some interesting results. Basically, in his analytically continued zeta-function, taking in complex values, he says that all the non trivial zeros have a real part one-half. It's difficult to comprehend why this is significant, and what the significance of those values really are.
More interesting results for me was the expression of zeta in terms of a product solely involving an expression of primes,
and the analytic continuation of the zeta function, which you can generate by taking the more-likely-to-converge eta function and manipulating a bit,
or taking this, which I have no idea how to derive from anything:
Also the gamma function is mind-blowing....
So apart from the math that I still don't really understand, I got a pretty good survey of the interesting historical figures that have populated math since Euler, including Gauss, Dirichlet, Riemann himself, Dedekind, Hilbert, (these five excluding Euler surrounded by the quirky atmosphere of Gottingen which I imagined as a black gothic fort), i forgot what Poussin Hadamard Chebyshev and Landau did, and then of course you have math's Tom and Jerry - Hardy and littlewood, who proved that there were a infinite number of non-trivial zeroes and that Li(x) intersects with pi(n) an infinite number of times (which was very surprising), respectively.
______________Rant______________
1. A big deal was made about big oh of the error term, ie the error term would not exceed modulus x^1/2 logx but okay... I thought Riemann already made a precise formulation of pi(x), given the truth of the hypotheses so... isn't this just a far weaker statement... I did not understand this section at all including the Mobius Mu stuff. So M(k) = O(k^1/2) ????? the inverse of zeta s is equivalent to sqrt(k)??? what does that even mean....??? :(
(Apparently this is a way of saying that if the zeta zeroes don't misbehave, the error term does not misbehave, which was not explicit in Riemann's formulations)
2. So at least we know that the primes are distributed as randomly as one would expect of a Normal or Gaussian distribution and that this may somehow be related to GUEs in QM, which are Hermitian matrices with coeffecients plucked randomly from a normal distribution. But what's the relation??? Idk :(
3. Then there's J(x), which assigns a weighted value to pi of the roots of x. ie
It wasn't explained how or why these values were weighted.
So you can forumlate zeta in terms of this weird shit in terms of J(x)
Idk why it's so important that you'd wanna see zeta in terms of this sort of integral but okay...
4. And fina-fuckin-lly we return to Riemann who manages to formulate J(x) in terms of the zeta function (I thought we already did that????)(and how exactly do you do that?????
So now we have a way of formulating pi(x) in terms of J(x) in terms of zeta (by means of the non-trivial zeta zeroes, in terms of sum of x^all the roots). So we have this really mysterious relation between the values of the zeroes and the more significant "error term" of J(x). Li(x^row) gives you a spiral that winds around pi(the real transcendental number)i and -pi(i)(why pi?) and SOMEHOW this ALL MANAGES TO WORK OUT?
______________________________________
And thus we get a super complicated way of counting the primes (and I sock the guy who says it's elegant, cos wtf is this beast of a thing?).
The zeta zeroes seem just as complicated as the distribution of primes themselves (could one say equivalently complicated), so one would wonder why you would represent one intractable problem in terms of another.
Maybe they're just far too random and unlike most things in the universe of mathematics, don't bear the special transcendental hand print of god. Nevertheless, we choose how to bide our time, and maybe playing games with numbers is just the same as anything else...
December 30, 2014
I read the book somewhere, i don't remember on whose laptop, but I was more than halfway through and the book made me feel really great. Its a very well written book. You need not to know much mathematics to start reading it, he teaches you along the way. And then he takes you from history to rigorous mathematics and that's awesome.
Certainly its one of the books out there in world - to enlighten!
Certainly its one of the books out there in world - to enlighten!
June 7, 2016
Prime Obsession is an engrossing and mind stretching journey to the heart of one of the most enduring and profound mysteries in mathematics - the Riemann Hypothesis:
All non-trivial zeros of the zeta function have real part one-half.
By the time you finish the book, that enigmatic statement along with the math behind it will make sense,you will have a deep understanding of the significance of TRH (namely how it is connected to the distribution of prime numbers) and you will have a feel for the rich history surrounding the hypothesis. All in all, Prime Obsession is a damn fine read and one I recommend to any mathematically curious person.
All non-trivial zeros of the zeta function have real part one-half.
By the time you finish the book, that enigmatic statement along with the math behind it will make sense,you will have a deep understanding of the significance of TRH (namely how it is connected to the distribution of prime numbers) and you will have a feel for the rich history surrounding the hypothesis. All in all, Prime Obsession is a damn fine read and one I recommend to any mathematically curious person.
August 18, 2018
If you are interested in mathematics in general, enjoy recreational mathematics or have even a curiosity and affinity for numbers then I think you’ll enjoy this book. It’s ‘popular�, in that it’s aimed at non specialists but there are minimum requirements for you to make the most of it. I think in the UK you probably need advanced level maths, or in the US have included some calculus at high school or early college. My background is the advanced maths that go with non-theoretical physics degrees and the complex functions associated with elasticity theory (in mechanical engineering). Even so, I found it required a good degree of concentration in the latter sections of the book!
Formally the author states he’s intending to explain the Riemann Hypothesis; “All non trivial zeros of the zeta function have real part 1/2�. That probably doesn’t mean much to you, as it didn’t to me either. But the journey to understanding what that means, and why it’s important in understanding how prime numbers are distributed amongst all numbers, is the mission of the book.
The author is very good at reintroducing mathematical topics you’ll need to understand his ultimate aims concerning Prime Numbers and the functions to model their distribution. So he gradually goes through basic definitions of numbers (real, integers, transcendental, etc); important functions for later work (ln x, and e^x); infinite series, converging and diverging, and order dependent; elementary calculus; complex numbers. It’s probably a bit much to be learning these for the first time via the book. I saw these as useful refreshers.
He spends time, of course, on prime numbers, and how empirically they are distributed amongst other numbers, before moving onto the important modelling of their distribution. The functions used to model how prime numbers are distributed and differences to the real distribution of numbers is getting us to the meat of the book.
The zeta function is introduced, examined, and importantly the way it can be reconstructed as other (geometrical) series. The link between ‘zeros� of the function and prime numbers is introduced, thus meeting the first part of this book’s mission.
I started to concentrate seriously when the zeta function was ‘mapped� to allow you to understand what these solutions mean. My reading rate slowed and I deliberately read in short spells so that I could understand bit by bit.
After this the discussion is a little less targeted, mainly because there seems to be a lack of momentum to proving it. Some interesting general discussion on why prime numbers may follow the apparently statistical functions that model their distribution.
Coming from a practical technical background I sometimes asked if the full blown theoretical proof mattered much given that the hypothesis seems valid numerically, as shown by computers, to numbers beyond practical use. But the possible links between prime numbers and various natural phenomena (certainly some aspects of atomic theory, and possibly sub-atomic particle physics - which was why I read up on this subject) mean that a theoretical proof would reinforce the importance of the physical links if demonstrated.
In addition, a bit of work by Littlewood in the 1920s proving that the difference between functions modelling prime numbers, and the actual distribution, does some strange things at uncountable large numbers was interesting. And maybe a reason that the theory could be false theoretically while apparently valid for practically useful numbers.
Let me also mention the book structure., which was effective for me. The heavy duty mathematics was dealt with in alternate chapters, with the intermediate chapters, as light relief, discussing background subjects such as the mathematicians doing the signifcant work (Euler, Gauss, Riemann, and more modern mathematicians). OK, they are not normally a bundle of laughs but even learning how they applied themselves to their life’s work was good to discover, and did make the book as a whole easier to read.
All in all, I think the author did an excellent job in explaining a difficult subject. I’m glad I persevered. Though again with the caveat that it’s not really for absolute beginners to the sort of maths one covers in the final stages of school, or 1st year technical degrees.
Formally the author states he’s intending to explain the Riemann Hypothesis; “All non trivial zeros of the zeta function have real part 1/2�. That probably doesn’t mean much to you, as it didn’t to me either. But the journey to understanding what that means, and why it’s important in understanding how prime numbers are distributed amongst all numbers, is the mission of the book.
The author is very good at reintroducing mathematical topics you’ll need to understand his ultimate aims concerning Prime Numbers and the functions to model their distribution. So he gradually goes through basic definitions of numbers (real, integers, transcendental, etc); important functions for later work (ln x, and e^x); infinite series, converging and diverging, and order dependent; elementary calculus; complex numbers. It’s probably a bit much to be learning these for the first time via the book. I saw these as useful refreshers.
He spends time, of course, on prime numbers, and how empirically they are distributed amongst other numbers, before moving onto the important modelling of their distribution. The functions used to model how prime numbers are distributed and differences to the real distribution of numbers is getting us to the meat of the book.
The zeta function is introduced, examined, and importantly the way it can be reconstructed as other (geometrical) series. The link between ‘zeros� of the function and prime numbers is introduced, thus meeting the first part of this book’s mission.
I started to concentrate seriously when the zeta function was ‘mapped� to allow you to understand what these solutions mean. My reading rate slowed and I deliberately read in short spells so that I could understand bit by bit.
After this the discussion is a little less targeted, mainly because there seems to be a lack of momentum to proving it. Some interesting general discussion on why prime numbers may follow the apparently statistical functions that model their distribution.
Coming from a practical technical background I sometimes asked if the full blown theoretical proof mattered much given that the hypothesis seems valid numerically, as shown by computers, to numbers beyond practical use. But the possible links between prime numbers and various natural phenomena (certainly some aspects of atomic theory, and possibly sub-atomic particle physics - which was why I read up on this subject) mean that a theoretical proof would reinforce the importance of the physical links if demonstrated.
In addition, a bit of work by Littlewood in the 1920s proving that the difference between functions modelling prime numbers, and the actual distribution, does some strange things at uncountable large numbers was interesting. And maybe a reason that the theory could be false theoretically while apparently valid for practically useful numbers.
Let me also mention the book structure., which was effective for me. The heavy duty mathematics was dealt with in alternate chapters, with the intermediate chapters, as light relief, discussing background subjects such as the mathematicians doing the signifcant work (Euler, Gauss, Riemann, and more modern mathematicians). OK, they are not normally a bundle of laughs but even learning how they applied themselves to their life’s work was good to discover, and did make the book as a whole easier to read.
All in all, I think the author did an excellent job in explaining a difficult subject. I’m glad I persevered. Though again with the caveat that it’s not really for absolute beginners to the sort of maths one covers in the final stages of school, or 1st year technical degrees.
May 3, 2009
This comparison will probably strike most as directly from left field, but Derbyshire reminds me a lot of . Topically, of course, they have nothing in common. But their style both depends heavily on the conspicuousness of the author in the narrative. This isn't necessarily because Krakauer and Derbyshire are narcissistic or self-absorbed, but that their writing is very self-conscious and they feel a continual impetus to advise the reader of where they stand on the issues they are presenting.
In some ways, these two are more journalists than they are chroniclers of history. And I suspect that neither would be more uncomfortable with that characterization. But something about this self-aware presentism has always bothered me. It bothers me deeply in Krakauer's work -- which is somehow concerned with what it means to be a flawed human animal in a time that mistrusts impulse and animalism; and though it bothers me somewhat less in Derbyshire -- who is dealing with the mirror aspect of humanity -- it still bothers me.
I guess the truth is that I would rather have just had a really good story than have to also hear what Derbyshire thought about it himself. Which is not to say there isn't a good story here! On the contrary. It's quite an excellent story! This book deserves at least the half a star above three that I'm forbidden from giving it.
In some ways, these two are more journalists than they are chroniclers of history. And I suspect that neither would be more uncomfortable with that characterization. But something about this self-aware presentism has always bothered me. It bothers me deeply in Krakauer's work -- which is somehow concerned with what it means to be a flawed human animal in a time that mistrusts impulse and animalism; and though it bothers me somewhat less in Derbyshire -- who is dealing with the mirror aspect of humanity -- it still bothers me.
I guess the truth is that I would rather have just had a really good story than have to also hear what Derbyshire thought about it himself. Which is not to say there isn't a good story here! On the contrary. It's quite an excellent story! This book deserves at least the half a star above three that I'm forbidden from giving it.
January 30, 2018
It's the Riemann hypothesis and the process of it.
It can't be solved yet 30 January 2018.
But the process of it is used various new mathematical method, they're exciting.
It can't be solved yet 30 January 2018.
But the process of it is used various new mathematical method, they're exciting.
July 17, 2011
This book is one of several books on a mathematical topic, ostensibly for laypersons. The topic in this case is the Riemann Hypothesis, which is one of the -- perhaps THE -- most important unsolved problems in Mathematics. The style and layout of the book follows one that I have seen in other such books, where the chapters alternate between the history and personalities and social and political context for those involved in trying to solve the problem, and an explanation of the mathematical topic or related topics.
In the case of the Riemann Hypothesis (RH), the mathematical explanations are not easy. The hypothesis itself is not easy to understand. The author defines the RH on page xi in the book's prologue: "The Riemann Hypothesis: All non-trivial zeros of the zeta function have real part one-half." He then spends a considerable part of the book explaining what that means. This involves introducing number systems, complex numbers, some integral calculus, and a lot of manipulation of infinite series.
I have a solid background in mathematics (my Ph.D. is in Elementary Particle Physics), albeit I'm on the rusty side, not having used my math much in recent years. I found some of the math explanations somewhat challenging, so I'm not sure what a genuine "intelligent layperson" would make of them. I found them interesting, and sometimes innovative, but towards the end I was growing tired and losing interest.
The chapters on the mathematicians, their history and social and political context were more interesting (and I am one who found history in high school to be boring in the extreme!)
On the whole, I liked the book, but I have also read Marcus du Sautoy's "The Music of the Primes", which is also focused on the RH, and I prefer that book.
Towards the end of this book, one starts to wonder what makes the RH so important to mathematicians. In Chapter 22, the author addresses the question "What use is it?", and then goes on to say that he, along with most pure mathematicians, don't really care whether there are applications or could be. But that is not the same as the question "Why is it important in mathematics?", and I felt after finishing the book that this latter question hadn't really been answered (unless you consider the entire contents of the book an answer; if so, it certainly isn't a succinct answer!)
In the case of the Riemann Hypothesis (RH), the mathematical explanations are not easy. The hypothesis itself is not easy to understand. The author defines the RH on page xi in the book's prologue: "The Riemann Hypothesis: All non-trivial zeros of the zeta function have real part one-half." He then spends a considerable part of the book explaining what that means. This involves introducing number systems, complex numbers, some integral calculus, and a lot of manipulation of infinite series.
I have a solid background in mathematics (my Ph.D. is in Elementary Particle Physics), albeit I'm on the rusty side, not having used my math much in recent years. I found some of the math explanations somewhat challenging, so I'm not sure what a genuine "intelligent layperson" would make of them. I found them interesting, and sometimes innovative, but towards the end I was growing tired and losing interest.
The chapters on the mathematicians, their history and social and political context were more interesting (and I am one who found history in high school to be boring in the extreme!)
On the whole, I liked the book, but I have also read Marcus du Sautoy's "The Music of the Primes", which is also focused on the RH, and I prefer that book.
Towards the end of this book, one starts to wonder what makes the RH so important to mathematicians. In Chapter 22, the author addresses the question "What use is it?", and then goes on to say that he, along with most pure mathematicians, don't really care whether there are applications or could be. But that is not the same as the question "Why is it important in mathematics?", and I felt after finishing the book that this latter question hadn't really been answered (unless you consider the entire contents of the book an answer; if so, it certainly isn't a succinct answer!)
July 9, 2007
Although I find this author's political views repellent, I really enjoyed this book. He takes an extremely esoteric mathematical puzzle and shows how it emerges organically starting from the simple math we learned in high school. He also provides several excellent character sketches of famous mathematicians who made the key discoveries that allowed the Riemann Hypothesis to come into being in the first place. Most importantly, Derbyshire manages to convey the sense that the field mathematics is a lively and active attempt to deduce new truths, not just a collection of proofs solved long ago.
July 20, 2018
I think this is the best pop science book I've ever read. For me, having been a math major in college but not studied or dealt with any high level math in probably around 15 years, it was pitched at a pretty good level. Most importantly, you can tell this was actually written by a professional mathematician rather than by a writer who is just dabbling.
There's a lot of handwaving around high level calculus but he explains what he's handwaving away enough that I was able to independently research most of it. One note though: although this book is (accurately!) billed as presenting the Reimann Hypothesis in the simplest, most understandable mathematics possible it's still pretty advanced. But I'd still recommend almost anybody to give it a go.
Ultimately, for the first time, I feel as if I genuinely understand the Reimann Hypothesis and why people care about it. Helping me understand arguably the most important unsolved problem in math is probably worth 5 stars.
There's a lot of handwaving around high level calculus but he explains what he's handwaving away enough that I was able to independently research most of it. One note though: although this book is (accurately!) billed as presenting the Reimann Hypothesis in the simplest, most understandable mathematics possible it's still pretty advanced. But I'd still recommend almost anybody to give it a go.
Ultimately, for the first time, I feel as if I genuinely understand the Reimann Hypothesis and why people care about it. Helping me understand arguably the most important unsolved problem in math is probably worth 5 stars.
January 29, 2013
Wonderful book for those who are interested in the subject and the modern math in general. The author has made his task very challengeable - to explain high level abstract math to a layman almost without even using calculus. And i have to say he succeeded! On the top of knowing the subject he is great storyteller. I wish I would have such a teacher for the Calculus when i was at the University;-)
It is not for everyone, but of you are seriously fascinated with math and have some knowledge in the subject I think you would enjoy it.
It is not for everyone, but of you are seriously fascinated with math and have some knowledge in the subject I think you would enjoy it.
January 19, 2008
This is a great book to get people interested in the prime number problems the last few thousand years. The first half of the book is very easy to read, and fun. The second half has more difficult mathematics in it.
May 24, 2012
I really enjoyed this book, but you have to take your time and work the math (which starts at really easy stuff, and is very carefully explained). You'll never look at prime number, or mathematics for that matter, in the same way again. -kevin aka FitOldDog
November 14, 2007
Try it. I dare you. (I bet you'll find it more interesting than you thought.)
January 30, 2014
WOuld have got 5, although I found the history chapters less interesting than the maths chapters. Definitely requires undergrad/advanced A level maths to follow but I really enjoyed it.
December 12, 2015
I'm such a math wannabe. Does a great job of breaking down a complicated hypothesis. Contained slightly too much people-history for my taste, but overall very good.
December 8, 2015
I know that in I never will fully understand the Riemann Hypothesis, not enough math cells in my brain, but this book have let me get a glimpse of the beauty of the zeta function.
December 19, 2022
The Riemann Hypothesis explained
was one of the greatest mathematicians of all time. Indeed, he is a strong candidate for GOAT. Mathematics, tells us, is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. is most famous for his work in analysis. Analysis is the mathematics of the continuum. In addition made contributions to geometry -- he was one of the discoverers of non-Euclidean geometry, which would eventually become the basis of 's Theory of General Relativity. And, most surprisingly, discovered a deep connection between analysis and arithmetic, number theory, the prime numbers. He set this out in one 1859 paper entitled "Über die Anzahl der Primzahlen unterhalb einer gegebenen Größe" -- "On the number of prime numbers below a given size". Within that paper he made a guess, which he couldn't prove, and wrote
What is the Riemann Hypothesis? It is a very technical conjecture about a mathematical object known as the Riemann zeta function. I cannot explain it briefly. At least half of is devoted to that purpose. You're not going to get a comprehensible shorter answer than by reading this book. I thoroughly enjoyed it.
Make no mistake -- this is a book about math. If you don't like math, you should probably not attempt it. However, it is also a book about mathematicians. The odd-numbered chapters cover the mathematics, while the even-numbered chapters cover history and biography. You could presumably read the even chapters alone to learn something about the Riemann Hypothesis while avoiding all the math. I didn't try that, so I can't tell you how well it would work.
is careful to describe himself as a journalist, not a mathematician. However, he obviously knows a great deal about mathematics and is good at explaining it. I read at the same time as my first reading of . is an attempt by , a card-carrying mathematician, to do what has done in . I was surprised to find that is much the better of the two books. It is not only that explains better -- his explanations are also, surprisingly, more mathematically rigorous than 's.
I was particularly impressed with 's handling of the theorem he grandly calls "The Golden Key", also known as Euler's product formula. He presents a complete and very clear proof. He explains how he found this proof as follows
Mathematicians are concerned, more than any other profession I know (including the arts) with the pursuit of beauty. Nonmathematicians are often surprised to hear this -- they don't perceive beauty in mathematics. has done an outstanding job of presenting mathematics as the beautiful thing it is.
.
was one of the greatest mathematicians of all time. Indeed, he is a strong candidate for GOAT. Mathematics, tells us, is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. is most famous for his work in analysis. Analysis is the mathematics of the continuum. In addition made contributions to geometry -- he was one of the discoverers of non-Euclidean geometry, which would eventually become the basis of 's Theory of General Relativity. And, most surprisingly, discovered a deep connection between analysis and arithmetic, number theory, the prime numbers. He set this out in one 1859 paper entitled "Über die Anzahl der Primzahlen unterhalb einer gegebenen Größe" -- "On the number of prime numbers below a given size". Within that paper he made a guess, which he couldn't prove, and wrote
One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation.Over 160 years later that guess, now known as the Riemann Hypothesis, remains unproved. writes, accurately I think, "The Riemann Hypothesis is now the great white whale of mathematical research."
What is the Riemann Hypothesis? It is a very technical conjecture about a mathematical object known as the Riemann zeta function. I cannot explain it briefly. At least half of is devoted to that purpose. You're not going to get a comprehensible shorter answer than by reading this book. I thoroughly enjoyed it.
Make no mistake -- this is a book about math. If you don't like math, you should probably not attempt it. However, it is also a book about mathematicians. The odd-numbered chapters cover the mathematics, while the even-numbered chapters cover history and biography. You could presumably read the even chapters alone to learn something about the Riemann Hypothesis while avoiding all the math. I didn't try that, so I can't tell you how well it would work.
is careful to describe himself as a journalist, not a mathematician. However, he obviously knows a great deal about mathematics and is good at explaining it. I read at the same time as my first reading of . is an attempt by , a card-carrying mathematician, to do what has done in . I was surprised to find that is much the better of the two books. It is not only that explains better -- his explanations are also, surprisingly, more mathematically rigorous than 's.
I was particularly impressed with 's handling of the theorem he grandly calls "The Golden Key", also known as Euler's product formula. He presents a complete and very clear proof. He explains how he found this proof as follows
When jotting down the ideas that make up this book, I first looked through some of the math texts on my shelves to find a proof of the Golden Key suitable for non-specialist readers. I settled on one that seemed to me acceptable and incorporated it. At a later stage of the book's development, I thought I had better carry out authorial due diligence, so I went to a research library (in this case the excellent new Science, Industry and Business branch of the New York Public Library in midtown Manhattan) and pulled out the original paper from Euler's collected works. His proof of the Golden Key covers ten lines and is far easier and more elegant than the one I had selected from my textbooks. I thereupon threw out my first choice of proof and replaced it with Euler's. The proof in part III of this chapter is essentially Euler's. It's a professorial cliché, I know, but it's true nonetheless: you can't beat going to the original sources.'s exposition of Euler's proof covers far more than ten lines -- Euler was writing for mathematicians and could abbreviate, knowing his readers would fill in the gaps. makes no such assumption and his proof is a thing of beauty.
Mathematicians are concerned, more than any other profession I know (including the arts) with the pursuit of beauty. Nonmathematicians are often surprised to hear this -- they don't perceive beauty in mathematics. has done an outstanding job of presenting mathematics as the beautiful thing it is.
.
July 6, 2019
Stoic and engaging attempt (with lots of history and sociology) to take one all the way through (almost from first principles) to understanding the near-mythical Riemann Hypothesis (RH). The most revelatory aspect of this exercise, to me, was the fact that even as a practicing "applied mathematician" (engineer) my knowledge of academic mathematics ends somewhere in the 18th century. Even the mathematics of Riemann from the middle of the 19th c. is too abstract to explain with simple numerical examples and plots and graphs. The book eventually gives up its doomed enterprise around that time-period and makes mathematical leaps (to be taken on faith) and manages to get past the finish line in explaining why it's possible that: "All non-trivial zeros of the zeta function have real part one-half"
December 28, 2015
A very good explanation of everything related to the Riemann Hypothesis for somebody not exposed to the rigorous maths of advanced level but enough to understand basic Calculus. The best thing about this book is that anybody with high school maths can just dive right in and follow the mathematics without much trouble.
The history presented along with the Math at some places is unnecessary but fun none the less.
The history presented along with the Math at some places is unnecessary but fun none the less.
April 3, 2017
This may be my favorite nonfiction mathematics book. The author alternates between biographical information on the main players in the history of the Riemann Hypothesis, and mathematical development of the background necessary to understand the hypothesis and its importance to modern mathematics. He manages to do this without requiring any advanced mathematics from his readers -- even calculus is used only once. Amazing, and well-written to boot.
August 16, 2014
Really nice mix of mathematical content and the historical story.
Seemed somewhat as though an editor favouring a lower page count rushed a dump of all remaining explanation in the last couple of chapters - dramatic increase in pace. Toward the end there are some odd mixes of assumed knowledge too.
Seemed somewhat as though an editor favouring a lower page count rushed a dump of all remaining explanation in the last couple of chapters - dramatic increase in pace. Toward the end there are some odd mixes of assumed knowledge too.
October 14, 2013
Беззаветно влюблена в Римана за его гениальный дзэн-шедевр, созерцать и наслаждаться которым можно до бесконечности. Нетривиальные нули правят миром, неважно верна гипотеза или нет! Кто бы мог подумать, что самая занудная с виду книга из серии полюбится больше всего
May 31, 2013
Great book with a historical perspective about the famous Hilbert's problem. A quick, enjoyable read to absorb the beauty of Number theory.
March 4, 2021
1. Harmonic series and intro to the different section of mathematics
a. Here we have a result: the harmonic series is divergent. How do you prove it? The proof is, in fact, rather easy and depends on nothing more than ordinary arithmetic. It was produced in the late Middle Ages by a French scholar, Nicole d'Oresme (ca. 1323-1382). D'Oresme pointed out that is greater than ; so is ; so is ; and so on. In other words, by taking 2 terms, then 4 terms, then 8, then 16 terms, and so on, you can group the series into an infinite number of blocks, every one of which is bigger than one-half. The entire sum must, therefore, be infinite.
b. The 4 main areas of mathematics: arithmetic, the study of whole numbers and fractions; geometry, the study of figures in space; algebra, the use of abstract symbols to represent mathematical objects; and analysis, the study of limits. The traditional categories have also been enlarged to include big new topics—geometry to include topology, algebra to take in game theory, and so on. Even before the early nineteenth century there was considerable seepage from one area into another. Trigonometry, for example, (the word was first used in 1595) contains elements of both geometry and algebra. Descartes had in fact arithmetized and algebraized a large part of geometry in the seventeenth century, though pure-geometric demonstrations in the style of Euclid were still popular—and still are—for their clarity, elegance, and ingenuity. The fourfold division is still a good rough guide to finding your way around mathematics, though.
c. the yoking of arithmetic to analysis to create an entirely new field of study, analytic number theory. Permit me to introduce the man who, with one single published paper of eight and a half pages, got analytic number theory off the ground and flying:Riemann
2. The PNT (prime number theorem)
a. Conjecture by gauss by studying his prime number tables. He send a letter to encke "i soon perceived that beneth all of its fluctuation, this frequency is, on average, close to inversely proportional with the logarithm.
b. Also published by legendre in his book titled essay on the theory of numbers before gauss wrote about was twiddle x/ logx - A "where a tended to some number near 1.08366, discussed as a false value in another of gauss's letters.
c. Then looking at it from a statistical point of view if we say that there are n/logn primes from 1 to n then the probability of a number being prime is twiddle 1/logn
d. And the pnt tells us that the nth prime number is twiddle nlogn.
3. Riemanns zeta function
a. The basel problem: to find a closed form for the series of reciprocal squares. The problem was finally cracked in 1735, 46 years after being posed, by the young Leonhard Euler, toiling away in St. Petersburg. Found to b pi squared divided by 2. His method gave the answer limit of the zeta function to any even number.
b. Random note: even though log x is not equal to x0, it nonetheless manages to dip below, and stay below, x , for any number , no matter how tiny, when x is large enough.27 The matter is, in fact, even stranger than that. Consider this statement: “The function log x eventually increases more slowly than x0.001, or x0.00001, or x0.0000001, or�.� Suppose I raise this whole statement to some power—say, the hundredth power, it will still increase at a slower rate
c.
d. Random note: Take any two numbers with no common factor and repetitively add one to the other. You will generate an infinity of primes (mixed with an infinity of non-primes). Gauss had conjectured that this was the case—knowing Gauss's powers, one is tempted to say that he intuited it—but it was decisively proved by Dirichlet in that 1837 paper.
e. The zeta function can be rearranged to
f.
4. An update of the pnt
a. The integral of 1/logt can be rewritten as li(x)
as N gets larger, Li(N) ~ N / log N. Now, the PNT asserts that (N) ~ N / log N. A moment's thought will convince you that the twiddle sign is transitive—that is, if P ~ Q and Q ~ R, then it must be the case that P ~ R. So if the PNT is true—which we know it is, it was proved in 1896—then it must also be true that (N)~Li(N).
b. Li(N) is actually a better estimate of (N) than N / log N is. A much better estimate.
c. El segundo paper de chebyschev enseño el proof of “Bertrand's postulate,� suggested in 1845 by the French mathematician Joseph Bertrand. The postulate states that between any number and its double (for example, between 42 and 84) there is always a prime to be found. The second was the one shown here. pi(N) cannot differ from n/logN by more than about ten percent up or down.
5. Rieman siendo to brainy y un poquitin de su famous paper on the number of primes less than a given quantity
a. Para unirse a no se que tuvo que escirbir un thesis y prepara un trial lecture. Gauss picked the lecture titled “On the Hypotheses that Lie at the Foundations of Geometry,� and Riemann delivered it to the assembled faculty on June 10, 1854. This is one of the top 10 mathematical papers ever delivered anywhere, a sensational achievement. Its reading was, declares Hans Freudenthal in the Dictionary of Scientific Biography, “one of the highlights in the history of mathematics.� The ideas contained in this paper were so advanced that it was decades before they became fully accepted, and 60 years before they found their natural physical application, as the mathematical framework for Einstein's General Theory of Relativity. James R. Newman, in The World of Mathematics, refers to the paper as “epoch-making� and “imperishable� (but fails to include it in his huge anthology of classic mathematical texts). And the astonishing thing is that the paper contains almost no mathematical symbolism. Leafing through it, I see five equals signs, three square root signs and four signs—an average of fewer than one symbol per page! There is just one real formula. The whole thing was written to be understood—or perhaps (see below) misunderstood—by the average faculty member of a middling provincial university.
b. The function of a complex variable s which both these expressions stand for, so long as they converge, i signify by [zeta] sign.
c. The non trivial zeros are all the negative even numbers as they always converge when put in the zeta function
d. This is the function; “� is “eta,� the seventh letter of the Greek alphabet, and I define the eta function as In a rough sort of way, you can see that this has a better prospect of converging than Expression 9-1.
Tien la pinta de 1 -1/(2^s) + 1/(3^s) � Instead of relentlessly adding numbers, we are alternately adding, then subtracting, so each number will to some extent cancel out the effect of the previous number. So it happens. Mathematicians can prove, in fact—though I'm not going to prove it here—that this new infinite series converges whenever s is greater than zero. This is a big improvement on Expression 9-1, which converges only for s greater than 1. What use is that for telling us anything about the zeta function? Well, first note the elementary fact of algebra that A � B + C � D + E � F + G � H + � is equal to (A + B + C + D + E + F + G + H + �) minus 2 × (B + D + F + H + �). So I can rewrite (s) as (la formula orginal todo positivo minus 2*(los reciprocals de los numbers pares) The first parenthesis is of course just (zeta function). The second parenthesis can be simplified by Power Rule 7, (ab)n = anbn. So every one of those even numbers can be broken up like this: , and I can take out as a factor of the whole parenthesis. Leaving what inside the parenthesis? Leaving (zeta)! In a nutshell
Zetafunction equals the eta function divided by 1 - reciprocal of 2 to the power of s-1 Now, this means that if I can figure out a value for (s), then I can easily figure out a value for (s). And since I can figure out values for (s) between 0 and 1, I can get a value for (s) in that range, too, in spite of the fact that the “official� series for (s) (Expression 9-1) doesn't converge there.
e. One of the results in Riemann's 1859 paper proves a formula first suggested by Euler in 1749, giving (1 � s) in terms of (s). So if you want to know the value of, say, (�15), you can just calculate (16) and feed it into the formula. It's a heck of a formula, though, and I give it here just for the sake of completeness. If you find that a little over the top, just take it on faith that there is a way to get a value of (s) for any number s, with the single exception of s = 1. Even if that last formula bounces right off your eye, at least notice this: it gives (1 � s) in terms of (s). That means that if you know (16) you can calculate (�15
f. Convergent series fall into two categories: those that have this property, and those that don't. Series like this one, whose limit depends on the order in which they are summed, are called “conditionally convergent.� Better-behaved series, those that converge to the same limit no matter how they are rearranged, are called “absolutely convergent.�
g. The PNT follows from a much weaker result (which has no name): All non-trivial zeros of the zeta function have real part less than one. If you can prove this, then you can use von Mangoldt's 1895 version of Riemann's main result to prove the PNT. That is what our two scholars did in 1896.
6. Intro into complex numbers and their properties and matrices
a. By Pythagoras's Theorem, the modulus of a + bi is . It is always a positive number or zero. The amplitude of a complex number is the angle it makes with the positive real line, measured in radians. (One radian is 57.29577951308232� degrees; 180 degrees is radians.) The amplitude is conventionally taken to be an angle between � (exclusive) and (inclusive) radians, and its symbol is Am(z).
b. Finally, the complex conjugate of a complex number is its mirror image in the real line. The complex conjugate of a + bi is a � bi. Its symbol is , pronounced “z bar.� If you multiply a complex number by its conjugate, you get a real number: (a + bi) × (a � bi) = a2 + b2, which is, in fact, the modulus of a + bi, squared.
7. The locationo of the non trivial zeros
a. As soon as riemanns hypothesis has been successfully established, the next problem would consist in testing more exactly Riemann's infinite series for the number of primes below a given number and, especially, to decide whether the difference between the number of primes below a number x and the integral logarithm of x does in fact become infinite of an order not greater than in x. Further, we should determine whether the occasional condensation of prime numbers which has been noticed in counting primes is really due to those terms of Riemann's formula which depend upon the first complex zeros of the function (s).
b. What was known up to 1900 about the zeros There is an infinity of them, all having real parts between 0 and 1 (exclusive). Using the complex plane to visualize this (see Figure 12-1), mathematicians say that all non-trivial zeros are known to lie in the critical strip. The Riemann Hypothesis makes a much stronger assertion, that they all lie on the line whose real part is one-half, that is, on the critical line. “Critical strip� and “critical line� are common terms of art in discussions of the Riemann Hypothesis, and from now on I shall use them quite freely. The Riemann Hypothesis (stated geometrically) All non-trivial zeros of the zeta function lie on the critical line. The zeros occur in conjugate pairs. That is, if a + bi is a zero, then so is a � bi. In other words, if z is a zero, then so is its complex conjugate . I defined “complex conjugate� and the z-bar notation in Chapter 11.v. In yet other words, if there is a zero above the real line, its mirror image below the real line is also a zero (and, of course, vice versa). Their real parts are symmetrical about the critical line; that is, a zero either has real part equal to (in line with the Hypothesis), or is one of a pair with real parts and , for some real number between 0 and , and identical imaginary parts. Real parts 0.43 and 0.57 are an example, or real parts 0.2 and 0.8. Another way of saying this would be: supposing there is any non-trivial zero not on the critical line, its mirror image in the critical line must also be a zero. This follows from that formula in Chapter 9.vi. If one side of that formula is zero, the other side must be too. Leaving aside integer values of s, where other terms in the formula misbehave or go to zero, this formula says that if (s) is zero, then (1 –s) must be zero too. Thus, if is a zero of the zeta function, then so is , and so, by the previous bullet point, is the conjugate .
c. Visualizing the transformation by a function on a complex plane: Take the entire complex plane. Make a cut along the negative real (west) axis, stopping at the zero point. Now grab the top half of that cut and pull it round counter-clockwise, using the zero point as a hinge. Stretch it right round through 360 degrees. Now it's over the stretched sheet, with the other side of the cut under the sheet. Pass it through the sheet (you have to imagine that the complex plane is not only infinitely stretchable, but also is made of a sort of misty substance that can pass through itself) and rejoin the original cut. Your mental picture now looks something like Figure 13-3. That is what the squaring function does to the complex plane.
d. Este es el argument plane del zeta function que enseño los punton que zeeta sends to the real and imaginary axes
e. There is in fact a rule for the average spacing of zeros at height T in the critical strip. It is ~ 2 / log(T / 2 ).
Now suppose that the argument ant, instead of following those fancy loops and whorls in Figure 13-6 (which send the value ant on dull hikes up and down the real and imaginary axes), takes a walk straight up the critical line, heading due north from argument . What path will the value ant follow? Figure 13-8 shows you. His path starts out at , which, as I showed in Chapter 9.v, is �1.4603545088095�. Then he does a sort of half-circle counter-clockwise below the zero point, then turns and loops clockwise around 1. He heads to zero and passes through it (that's the first zero—the argument ant has just passed ). Then he keeps going round in clockwise loops, passing through the zero point every so often—whenever his twin on the argument plane steps on a zero of the zeta function. I stopped his walk when the argument ant reached , because that's as far as Figure 13-6 goes. By that point, the curve has passed through zero five times, corresponding to the five non-trivial zeros in Figure 13-6. Notice that points on the critical line have a strong tendency to map to points with positive real part
f. Hardy paper on the the zeros proves that infinetly many of the zeta funciton's non trivial zeros satisfy therie
a. Here we have a result: the harmonic series is divergent. How do you prove it? The proof is, in fact, rather easy and depends on nothing more than ordinary arithmetic. It was produced in the late Middle Ages by a French scholar, Nicole d'Oresme (ca. 1323-1382). D'Oresme pointed out that is greater than ; so is ; so is ; and so on. In other words, by taking 2 terms, then 4 terms, then 8, then 16 terms, and so on, you can group the series into an infinite number of blocks, every one of which is bigger than one-half. The entire sum must, therefore, be infinite.
b. The 4 main areas of mathematics: arithmetic, the study of whole numbers and fractions; geometry, the study of figures in space; algebra, the use of abstract symbols to represent mathematical objects; and analysis, the study of limits. The traditional categories have also been enlarged to include big new topics—geometry to include topology, algebra to take in game theory, and so on. Even before the early nineteenth century there was considerable seepage from one area into another. Trigonometry, for example, (the word was first used in 1595) contains elements of both geometry and algebra. Descartes had in fact arithmetized and algebraized a large part of geometry in the seventeenth century, though pure-geometric demonstrations in the style of Euclid were still popular—and still are—for their clarity, elegance, and ingenuity. The fourfold division is still a good rough guide to finding your way around mathematics, though.
c. the yoking of arithmetic to analysis to create an entirely new field of study, analytic number theory. Permit me to introduce the man who, with one single published paper of eight and a half pages, got analytic number theory off the ground and flying:Riemann
2. The PNT (prime number theorem)
a. Conjecture by gauss by studying his prime number tables. He send a letter to encke "i soon perceived that beneth all of its fluctuation, this frequency is, on average, close to inversely proportional with the logarithm.
b. Also published by legendre in his book titled essay on the theory of numbers before gauss wrote about was twiddle x/ logx - A "where a tended to some number near 1.08366, discussed as a false value in another of gauss's letters.
c. Then looking at it from a statistical point of view if we say that there are n/logn primes from 1 to n then the probability of a number being prime is twiddle 1/logn
d. And the pnt tells us that the nth prime number is twiddle nlogn.
3. Riemanns zeta function
a. The basel problem: to find a closed form for the series of reciprocal squares. The problem was finally cracked in 1735, 46 years after being posed, by the young Leonhard Euler, toiling away in St. Petersburg. Found to b pi squared divided by 2. His method gave the answer limit of the zeta function to any even number.
b. Random note: even though log x is not equal to x0, it nonetheless manages to dip below, and stay below, x , for any number , no matter how tiny, when x is large enough.27 The matter is, in fact, even stranger than that. Consider this statement: “The function log x eventually increases more slowly than x0.001, or x0.00001, or x0.0000001, or�.� Suppose I raise this whole statement to some power—say, the hundredth power, it will still increase at a slower rate
c.
d. Random note: Take any two numbers with no common factor and repetitively add one to the other. You will generate an infinity of primes (mixed with an infinity of non-primes). Gauss had conjectured that this was the case—knowing Gauss's powers, one is tempted to say that he intuited it—but it was decisively proved by Dirichlet in that 1837 paper.
e. The zeta function can be rearranged to
f.
4. An update of the pnt
a. The integral of 1/logt can be rewritten as li(x)
as N gets larger, Li(N) ~ N / log N. Now, the PNT asserts that (N) ~ N / log N. A moment's thought will convince you that the twiddle sign is transitive—that is, if P ~ Q and Q ~ R, then it must be the case that P ~ R. So if the PNT is true—which we know it is, it was proved in 1896—then it must also be true that (N)~Li(N).
b. Li(N) is actually a better estimate of (N) than N / log N is. A much better estimate.
c. El segundo paper de chebyschev enseño el proof of “Bertrand's postulate,� suggested in 1845 by the French mathematician Joseph Bertrand. The postulate states that between any number and its double (for example, between 42 and 84) there is always a prime to be found. The second was the one shown here. pi(N) cannot differ from n/logN by more than about ten percent up or down.
5. Rieman siendo to brainy y un poquitin de su famous paper on the number of primes less than a given quantity
a. Para unirse a no se que tuvo que escirbir un thesis y prepara un trial lecture. Gauss picked the lecture titled “On the Hypotheses that Lie at the Foundations of Geometry,� and Riemann delivered it to the assembled faculty on June 10, 1854. This is one of the top 10 mathematical papers ever delivered anywhere, a sensational achievement. Its reading was, declares Hans Freudenthal in the Dictionary of Scientific Biography, “one of the highlights in the history of mathematics.� The ideas contained in this paper were so advanced that it was decades before they became fully accepted, and 60 years before they found their natural physical application, as the mathematical framework for Einstein's General Theory of Relativity. James R. Newman, in The World of Mathematics, refers to the paper as “epoch-making� and “imperishable� (but fails to include it in his huge anthology of classic mathematical texts). And the astonishing thing is that the paper contains almost no mathematical symbolism. Leafing through it, I see five equals signs, three square root signs and four signs—an average of fewer than one symbol per page! There is just one real formula. The whole thing was written to be understood—or perhaps (see below) misunderstood—by the average faculty member of a middling provincial university.
b. The function of a complex variable s which both these expressions stand for, so long as they converge, i signify by [zeta] sign.
c. The non trivial zeros are all the negative even numbers as they always converge when put in the zeta function
d. This is the function; “� is “eta,� the seventh letter of the Greek alphabet, and I define the eta function as In a rough sort of way, you can see that this has a better prospect of converging than Expression 9-1.
Tien la pinta de 1 -1/(2^s) + 1/(3^s) � Instead of relentlessly adding numbers, we are alternately adding, then subtracting, so each number will to some extent cancel out the effect of the previous number. So it happens. Mathematicians can prove, in fact—though I'm not going to prove it here—that this new infinite series converges whenever s is greater than zero. This is a big improvement on Expression 9-1, which converges only for s greater than 1. What use is that for telling us anything about the zeta function? Well, first note the elementary fact of algebra that A � B + C � D + E � F + G � H + � is equal to (A + B + C + D + E + F + G + H + �) minus 2 × (B + D + F + H + �). So I can rewrite (s) as (la formula orginal todo positivo minus 2*(los reciprocals de los numbers pares) The first parenthesis is of course just (zeta function). The second parenthesis can be simplified by Power Rule 7, (ab)n = anbn. So every one of those even numbers can be broken up like this: , and I can take out as a factor of the whole parenthesis. Leaving what inside the parenthesis? Leaving (zeta)! In a nutshell
Zetafunction equals the eta function divided by 1 - reciprocal of 2 to the power of s-1 Now, this means that if I can figure out a value for (s), then I can easily figure out a value for (s). And since I can figure out values for (s) between 0 and 1, I can get a value for (s) in that range, too, in spite of the fact that the “official� series for (s) (Expression 9-1) doesn't converge there.
e. One of the results in Riemann's 1859 paper proves a formula first suggested by Euler in 1749, giving (1 � s) in terms of (s). So if you want to know the value of, say, (�15), you can just calculate (16) and feed it into the formula. It's a heck of a formula, though, and I give it here just for the sake of completeness. If you find that a little over the top, just take it on faith that there is a way to get a value of (s) for any number s, with the single exception of s = 1. Even if that last formula bounces right off your eye, at least notice this: it gives (1 � s) in terms of (s). That means that if you know (16) you can calculate (�15
f. Convergent series fall into two categories: those that have this property, and those that don't. Series like this one, whose limit depends on the order in which they are summed, are called “conditionally convergent.� Better-behaved series, those that converge to the same limit no matter how they are rearranged, are called “absolutely convergent.�
g. The PNT follows from a much weaker result (which has no name): All non-trivial zeros of the zeta function have real part less than one. If you can prove this, then you can use von Mangoldt's 1895 version of Riemann's main result to prove the PNT. That is what our two scholars did in 1896.
6. Intro into complex numbers and their properties and matrices
a. By Pythagoras's Theorem, the modulus of a + bi is . It is always a positive number or zero. The amplitude of a complex number is the angle it makes with the positive real line, measured in radians. (One radian is 57.29577951308232� degrees; 180 degrees is radians.) The amplitude is conventionally taken to be an angle between � (exclusive) and (inclusive) radians, and its symbol is Am(z).
b. Finally, the complex conjugate of a complex number is its mirror image in the real line. The complex conjugate of a + bi is a � bi. Its symbol is , pronounced “z bar.� If you multiply a complex number by its conjugate, you get a real number: (a + bi) × (a � bi) = a2 + b2, which is, in fact, the modulus of a + bi, squared.
7. The locationo of the non trivial zeros
a. As soon as riemanns hypothesis has been successfully established, the next problem would consist in testing more exactly Riemann's infinite series for the number of primes below a given number and, especially, to decide whether the difference between the number of primes below a number x and the integral logarithm of x does in fact become infinite of an order not greater than in x. Further, we should determine whether the occasional condensation of prime numbers which has been noticed in counting primes is really due to those terms of Riemann's formula which depend upon the first complex zeros of the function (s).
b. What was known up to 1900 about the zeros There is an infinity of them, all having real parts between 0 and 1 (exclusive). Using the complex plane to visualize this (see Figure 12-1), mathematicians say that all non-trivial zeros are known to lie in the critical strip. The Riemann Hypothesis makes a much stronger assertion, that they all lie on the line whose real part is one-half, that is, on the critical line. “Critical strip� and “critical line� are common terms of art in discussions of the Riemann Hypothesis, and from now on I shall use them quite freely. The Riemann Hypothesis (stated geometrically) All non-trivial zeros of the zeta function lie on the critical line. The zeros occur in conjugate pairs. That is, if a + bi is a zero, then so is a � bi. In other words, if z is a zero, then so is its complex conjugate . I defined “complex conjugate� and the z-bar notation in Chapter 11.v. In yet other words, if there is a zero above the real line, its mirror image below the real line is also a zero (and, of course, vice versa). Their real parts are symmetrical about the critical line; that is, a zero either has real part equal to (in line with the Hypothesis), or is one of a pair with real parts and , for some real number between 0 and , and identical imaginary parts. Real parts 0.43 and 0.57 are an example, or real parts 0.2 and 0.8. Another way of saying this would be: supposing there is any non-trivial zero not on the critical line, its mirror image in the critical line must also be a zero. This follows from that formula in Chapter 9.vi. If one side of that formula is zero, the other side must be too. Leaving aside integer values of s, where other terms in the formula misbehave or go to zero, this formula says that if (s) is zero, then (1 –s) must be zero too. Thus, if is a zero of the zeta function, then so is , and so, by the previous bullet point, is the conjugate .
c. Visualizing the transformation by a function on a complex plane: Take the entire complex plane. Make a cut along the negative real (west) axis, stopping at the zero point. Now grab the top half of that cut and pull it round counter-clockwise, using the zero point as a hinge. Stretch it right round through 360 degrees. Now it's over the stretched sheet, with the other side of the cut under the sheet. Pass it through the sheet (you have to imagine that the complex plane is not only infinitely stretchable, but also is made of a sort of misty substance that can pass through itself) and rejoin the original cut. Your mental picture now looks something like Figure 13-3. That is what the squaring function does to the complex plane.
d. Este es el argument plane del zeta function que enseño los punton que zeeta sends to the real and imaginary axes
e. There is in fact a rule for the average spacing of zeros at height T in the critical strip. It is ~ 2 / log(T / 2 ).
Now suppose that the argument ant, instead of following those fancy loops and whorls in Figure 13-6 (which send the value ant on dull hikes up and down the real and imaginary axes), takes a walk straight up the critical line, heading due north from argument . What path will the value ant follow? Figure 13-8 shows you. His path starts out at , which, as I showed in Chapter 9.v, is �1.4603545088095�. Then he does a sort of half-circle counter-clockwise below the zero point, then turns and loops clockwise around 1. He heads to zero and passes through it (that's the first zero—the argument ant has just passed ). Then he keeps going round in clockwise loops, passing through the zero point every so often—whenever his twin on the argument plane steps on a zero of the zeta function. I stopped his walk when the argument ant reached , because that's as far as Figure 13-6 goes. By that point, the curve has passed through zero five times, corresponding to the five non-trivial zeros in Figure 13-6. Notice that points on the critical line have a strong tendency to map to points with positive real part
f. Hardy paper on the the zeros proves that infinetly many of the zeta funciton's non trivial zeros satisfy therie
dnf
May 29, 2022I bought this years ago. I began it but put it down. I picked it up last year and committed to finishing. It focusses on the work of Bernhard Riemann, specifically, the Riemann hypotheses which proposes an answer to the question about how to calculate the number of prime numbers less than some number X.
I am on page 252 and cannot go on.
I do not know why this theorem has exercised mathematicians for 150 years and Derbyshire does not help. I guess mostly what he does is go through mathematical theories that were developed to address the problem.
I am on page 252 and cannot go on.
I do not know why this theorem has exercised mathematicians for 150 years and Derbyshire does not help. I guess mostly what he does is go through mathematical theories that were developed to address the problem.
March 3, 2024
A really great and thorough account of the Riemann Hypothesis accessible to the layman, I do not rate it higher simply because of issues I personally found with pacing. It should be understood to be a high 3 stars. I'll have to try next!
December 21, 2020
Fabulous, John Derbyshire takes a complex topic and create a beautiful tapestry covering the people, history and status of research on the RH problem. An enjoyable and thought provoking read. Not to mention a great resource for citation material, and a nice aside - a song.
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