I love that this is a site solely dedicated to the subject of interesting Diophantine equations, written by two mathematics PhD students. Going deep on a narrow focus, I respect that approach.
The article itself feels a bit long without a satisfying payoff at end. But then again, the journey is entertaining even for a non-specialist, and the lack of a strong conclusion is due to the unsolved problem of the Riemann hypothesis. The open-ended question is probably irresistible for some personality types that can't stand the suspense and demand a resolution.
Perhaps a way to strengthen the ending is to explain why the question of the distribution of prime numbers is worth solving, and what are the larger implications.
I’ve seen the connection between the Riemann zeta function and primes talked about but never an accessible explanation. Although I must say that the beginning was fairly easy to follow, I got overwhelmed about a quarter into it. Feels like I need a week of study to really comprehend the entire article.
The simplest explanation would be the fact that the Riemann zeta function is also equal to the infinite product of 1/(1 - p^{-s}) for all primes p. The proof is rather accessible, see https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_for... . That’s sort of the simplest result that shows a relationship between primes and the zeta function. That’s what this article builds on, but doesn’t give that actual result until about a quarter of the way through.
I skimmed the article, but the next third of the article seems to be devoted to using this relationship between the zeta function and prime numbers to prove the prime number theorem, which is a theorem approximating how many primes are less than or equal to any given number N.
The final third goes into how to get increasingly accurate approximations for the number of primes less than N, ending on the fact that Gausses approximation is in some sense the “best”, but only if the Riemann zeta functions zeroes lie on the critical section.
If you just want a general primer on why the zeta function has anything to do with primes, the product formula might suffice. In which case, the proof on the Wikipedia page might be a better read. The derivation in the article focuses on the general setup that is later built on to prove additional things
Had same problem. One thing which is not always made clear: sound waves can be thought as a sum of sinusoidals of different frequencies (the Fourier Transform). The zeta functions does something similar: you add up the zeta functions for the non-trivial zeros, and the sum of them has jumps at the location of primes.
Highly simplified, and also wrong, you don't get the actual primes, but an error correcting term to another prime estimation function.
So in a way, the zeroes of the zeta function encode "the frequencies of the primes" (in the Fourier sense)
Apropos, Wikipedia says at a picture: "(Left) The von Mangoldt function, approximated by zeta zero waves.(Right) The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes." [1]. [2] explains the details very gently, one can skip parts. So this is a machine with input a function very close to prime counts, and outputs info about zeta zeros. And the machine is Fourier transform.
There is a whole YouTube channel just about this subject. Tens of hours in total, in 30-50 min episodes handling little chunks of this matter (Zeta/Riemann/primes)
Easy to follow without requiring advanced math, great visualizations.
However needing tens of hours of video to explain what the Riemann Hypothesis is, without glossing over details, tells you something about it's difficulty (as a statement).
I don't think tens of hours are necessary. There was a very good public talk by a mathematician from first principles that even a child can understand, all the way up to the Riemann hypothesis, in 50 minutes. I wish it was online. It was the best talk of any subject I have ever heard.
Love the writing style, and I wish all math was written this way. These articles are accessible to anyone who has had some high school math background, and paints a historical backdrop on how the sausage was made (with links to the original papers). Some of these ideas took millenia to develop, and when presented in neat capsule, students are cheated of the mystery and struggle that went into the progression and development of the ideas. I would argue that all math education has to be re-invented to follow this style, ie, uncover the historical progression, wonder and mystery.
Wholeheartedly agree that this is a great educational approach, telling the story of an idea's development through history, the people involved, the major milestones, how the original spark was shaped, questioned, extended, and refined to its present form. I often find myself tracing an idea "backward" through time, to discover where it came from, who was influenced by whom, and learn so much from the journey.
This story usually explains why some concept is the way it is, whereas the typical presentation is to just give you the final form, without untangling the logic and its historical development. Also, starting from the original conception - like ancient Greeks or Indian philosopher, etc. - is much more relatable because one can see the "primitive" shape of an idea. And I mean primitive in the best sense of the term, thinking from first principles.
Good article, but there is one step in the reasoning that rubs me the wrong way:
> This equation might seem a little hard to solve, but at this point you might notice something funny: $equation$
> So, if F′(x)log(x)=1,F′(x)log(x)=1, then
> Thus, our mystery function F(x)F(x) obeys F′(x)log(x)=1.F′(x)log(x)=1. From here you deduce F′(x)=1/log(x),F′(x)=1/log(x), so by integrating you get
But it does not seems that F needs to have that trait, just that 1 works in that instance. It's sufficient but not necessary. How can you tell that it is that simple solution which is the right one?
The Riemann Hypothesis is a weird thing. It is about as important as Fundamental Theorem of Algebra or Fundamental Theorem of Calculus in number theory. Yet we are nowhere near to proving it. And yet it is so powerful that we are trying to prove things assuming RH is correct.
I was curious why/how someone would even think about arranging numbers in a spiral. The origin story is funny:
> According to Martin Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of "a long and very boring paper" at a scientific meeting. These hand calculations amounted to "a few hundred points". Shortly afterwards, Ulam and collaborators used MANIAC II at Los Alamos Scientific Laboratory to extend the calculation to about 100,000 points.
Funnily enough, it was almost discovered several years earlier. The science fiction author Arthur C Clarke wrote in “The City and the Stars” a passage that, as an aside, describes a mathematician looking for patterns in the primes by arranging them in a spiral grid. But he never actually tried doing this himself, and so never actually saw the pattern.
From Chapter 6 of "The City and the Stars" (1953) by Arthur C Clarke.
JESERAC SAT MOTIONLESS within a whirlpool of numbers. The first thousand primes, expressed in the binary scale that had been used for all aritmetical operations since electronic computers were invented, marched in order before him. Endless ranks of 1's and 0's paraded past, bringing before Jeserac's eyes the complete sequence of all those numbers that possessed no factors except themselves and unity. There was a mystery about the primes that had always fascinated Man, and they held his imagination still.
Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed.
He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh. Jeserac had done this a hundred times before, and it had never taught him anything. But he was fascinated by the way in which the numbers he was studying were scattered, apparently according to no laws, across the spectrum of the integers. He knew the laws of distribution that had already been discovered, but always hoped to discover more.
He could scarcely complain about the interruption. If he had wished to remain undisturbed, he should have set his annunciator accordingly. As the gentle chime sounded in his ear, the wall of numbers shivered, the digits blurred together, and Jeserac returned to the world of mere reality.
I despise vague and quasi-anonymous attacks like this. "Trust me, online person, I, a random netizen, have made a moral judgment and you should have complete faith in it."
Google summarizes the controversy thus, from Wikipedia:
"John Derbyshire is an American journalist and political commentator. He was one of the last paleoconservatives at the National Review, until he was fired in 2012 for writing an article for Taki's Magazine that was widely described as racist. Since 2012 he has written for white nationalist website VDARE. Wikipedia
Is it not obvious? Replace “fairly despicable” with “white nationalist” instead of whatever anyone might imagine could make a mathematician despicable?
Personally I’ve found this article a much more approachable overview to the question in the title: https://golem.ph.utexas.edu/category/2019/09/the_riemann_hyp...
I love that this is a site solely dedicated to the subject of interesting Diophantine equations, written by two mathematics PhD students. Going deep on a narrow focus, I respect that approach.
The article itself feels a bit long without a satisfying payoff at end. But then again, the journey is entertaining even for a non-specialist, and the lack of a strong conclusion is due to the unsolved problem of the Riemann hypothesis. The open-ended question is probably irresistible for some personality types that can't stand the suspense and demand a resolution.
Perhaps a way to strengthen the ending is to explain why the question of the distribution of prime numbers is worth solving, and what are the larger implications.
I’ve seen the connection between the Riemann zeta function and primes talked about but never an accessible explanation. Although I must say that the beginning was fairly easy to follow, I got overwhelmed about a quarter into it. Feels like I need a week of study to really comprehend the entire article.
The simplest explanation would be the fact that the Riemann zeta function is also equal to the infinite product of 1/(1 - p^{-s}) for all primes p. The proof is rather accessible, see https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_for... . That’s sort of the simplest result that shows a relationship between primes and the zeta function. That’s what this article builds on, but doesn’t give that actual result until about a quarter of the way through.
I skimmed the article, but the next third of the article seems to be devoted to using this relationship between the zeta function and prime numbers to prove the prime number theorem, which is a theorem approximating how many primes are less than or equal to any given number N.
The final third goes into how to get increasingly accurate approximations for the number of primes less than N, ending on the fact that Gausses approximation is in some sense the “best”, but only if the Riemann zeta functions zeroes lie on the critical section.
If you just want a general primer on why the zeta function has anything to do with primes, the product formula might suffice. In which case, the proof on the Wikipedia page might be a better read. The derivation in the article focuses on the general setup that is later built on to prove additional things
Had same problem. One thing which is not always made clear: sound waves can be thought as a sum of sinusoidals of different frequencies (the Fourier Transform). The zeta functions does something similar: you add up the zeta functions for the non-trivial zeros, and the sum of them has jumps at the location of primes.
Highly simplified, and also wrong, you don't get the actual primes, but an error correcting term to another prime estimation function.
So in a way, the zeroes of the zeta function encode "the frequencies of the primes" (in the Fourier sense)
https://www.youtube.com/watch?v=aT0VbxAUwNA
Apropos, Wikipedia says at a picture: "(Left) The von Mangoldt function, approximated by zeta zero waves.(Right) The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes." [1]. [2] explains the details very gently, one can skip parts. So this is a machine with input a function very close to prime counts, and outputs info about zeta zeros. And the machine is Fourier transform.
[1] https://en.wikipedia.org/wiki/Von_Mangoldt_function
[2] Prime numbers and the Riemann hypothesis. Mazur, Stein
There is a whole YouTube channel just about this subject. Tens of hours in total, in 30-50 min episodes handling little chunks of this matter (Zeta/Riemann/primes)
Easy to follow without requiring advanced math, great visualizations.
https://www.youtube.com/@ZetaExplained/videos
However needing tens of hours of video to explain what the Riemann Hypothesis is, without glossing over details, tells you something about it's difficulty (as a statement).
I don't think tens of hours are necessary. There was a very good public talk by a mathematician from first principles that even a child can understand, all the way up to the Riemann hypothesis, in 50 minutes. I wish it was online. It was the best talk of any subject I have ever heard.
Love the writing style, and I wish all math was written this way. These articles are accessible to anyone who has had some high school math background, and paints a historical backdrop on how the sausage was made (with links to the original papers). Some of these ideas took millenia to develop, and when presented in neat capsule, students are cheated of the mystery and struggle that went into the progression and development of the ideas. I would argue that all math education has to be re-invented to follow this style, ie, uncover the historical progression, wonder and mystery.
Wholeheartedly agree that this is a great educational approach, telling the story of an idea's development through history, the people involved, the major milestones, how the original spark was shaped, questioned, extended, and refined to its present form. I often find myself tracing an idea "backward" through time, to discover where it came from, who was influenced by whom, and learn so much from the journey.
This story usually explains why some concept is the way it is, whereas the typical presentation is to just give you the final form, without untangling the logic and its historical development. Also, starting from the original conception - like ancient Greeks or Indian philosopher, etc. - is much more relatable because one can see the "primitive" shape of an idea. And I mean primitive in the best sense of the term, thinking from first principles.
Good article, but there is one step in the reasoning that rubs me the wrong way:
> This equation might seem a little hard to solve, but at this point you might notice something funny: $equation$
> So, if F′(x)log(x)=1,F′(x)log(x)=1, then
> Thus, our mystery function F(x)F(x) obeys F′(x)log(x)=1.F′(x)log(x)=1. From here you deduce F′(x)=1/log(x),F′(x)=1/log(x), so by integrating you get
But it does not seems that F needs to have that trait, just that 1 works in that instance. It's sufficient but not necessary. How can you tell that it is that simple solution which is the right one?
The Riemann Hypothesis is a weird thing. It is about as important as Fundamental Theorem of Algebra or Fundamental Theorem of Calculus in number theory. Yet we are nowhere near to proving it. And yet it is so powerful that we are trying to prove things assuming RH is correct.
And as if all of that were not head-exploding enough, I'm still searching for an implementation of the zeta function for complex arguments...
It doesn’t change if you apply it to complex arguments does it?
Zeta(z) = 1 + 1/2^z + 1/3^z + …
Where z in C.
In fact, I thought that was why it’s called the Riemann zeta function. Euler applied it to an integer whereas Riemann applied it to complex arguments.
Edit to add: my memory was correct. Reimann extended Euler’s definition to all complex s not equal to 1. https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_T...
that definition only converges for Re(z)>1. for Re(z)<=1, you need alternate formulas
Yes. Riemann gives them in the paper, including derivation, it’s just a bit annoying to type/read them here because hn doesn’t support mathjaxx.
https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-...
I'm just not mathematician enough to convert that into a C function. Let alone one that is both efficient and reasonably accurate.
Stupid question on toilet: if prime numbers are 1s and other integers are 0s. What do we get from this “digital” landscape?
If you plot the positive integers two-dimensionally in a square spiral arrangement, eg.
and so on, and mark all primes, what you get is the Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiralI was curious why/how someone would even think about arranging numbers in a spiral. The origin story is funny:
> According to Martin Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of "a long and very boring paper" at a scientific meeting. These hand calculations amounted to "a few hundred points". Shortly afterwards, Ulam and collaborators used MANIAC II at Los Alamos Scientific Laboratory to extend the calculation to about 100,000 points.
Funnily enough, it was almost discovered several years earlier. The science fiction author Arthur C Clarke wrote in “The City and the Stars” a passage that, as an aside, describes a mathematician looking for patterns in the primes by arranging them in a spiral grid. But he never actually tried doing this himself, and so never actually saw the pattern.
From Chapter 6 of "The City and the Stars" (1953) by Arthur C Clarke.
JESERAC SAT MOTIONLESS within a whirlpool of numbers. The first thousand primes, expressed in the binary scale that had been used for all aritmetical operations since electronic computers were invented, marched in order before him. Endless ranks of 1's and 0's paraded past, bringing before Jeserac's eyes the complete sequence of all those numbers that possessed no factors except themselves and unity. There was a mystery about the primes that had always fascinated Man, and they held his imagination still.
Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed.
He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh. Jeserac had done this a hundred times before, and it had never taught him anything. But he was fascinated by the way in which the numbers he was studying were scattered, apparently according to no laws, across the spectrum of the integers. He knew the laws of distribution that had already been discovered, but always hoped to discover more.
He could scarcely complain about the interruption. If he had wished to remain undisturbed, he should have set his annunciator accordingly. As the gentle chime sounded in his ear, the wall of numbers shivered, the digits blurred together, and Jeserac returned to the world of mere reality.
The first-order difference of the prime-counting function: https://en.wikipedia.org/wiki/Prime-counting_function
The decimal expansion of an irrational number https://oeis.org/A010051
Although the author turned out to be a fairly despicable person, Prime Obsession is an absolutely wonderful book on this subject.
This is one of the few places I ever engage on the internet, but I think it may be time for HN to be read-only, too.
This didn't seem especially controversial of a comment, and even here there's no end of the rotten attitudes for no good reason
What you call rotten attitudes seemed like fairly thoughtfully doing the research you left out of an opaque and off-topic character determination.
Read-only or read never is looking wiser than it did a few years ago.
I despise vague and quasi-anonymous attacks like this. "Trust me, online person, I, a random netizen, have made a moral judgment and you should have complete faith in it."
Google summarizes the controversy thus, from Wikipedia:
"John Derbyshire is an American journalist and political commentator. He was one of the last paleoconservatives at the National Review, until he was fired in 2012 for writing an article for Taki's Magazine that was widely described as racist. Since 2012 he has written for white nationalist website VDARE. Wikipedia
Would you prefer OC was more specific, ie “the author turned out to be a racist”, or do you have an issue with calling someone like this despicable?
Obviously, I would prefer OC was more specific. I said as much.
Genuine question: are you on the spectrum? I find this reaction strange
Wait... so you are perfectly OK with hating someone on the basis of some random online poster telling you that you should, with no explanation?
Wow.
I don’t automatically hate someone because someone calls them despicable
I always read that word as though it were being pronounced by Daffy Duck; "you're dethpikable!".
Is it not obvious? Replace “fairly despicable” with “white nationalist” instead of whatever anyone might imagine could make a mathematician despicable?
Why does this shorthand upset you?
A shorthand is generally unambiguous in context; this was not.
Why does it matter to you so much that OC is unambiguous?
It was off-topic already. You driving it further is obsessive and ill-intended.
Because I'm not on-board with every moral failing that is called “despicable” by someone being disqualifying or even relevant.
I am not upset. It is obviously obstusely, impenetrably vague. What is your problem?
What is the relationship between Riemann zeta function and Zipf's law? And between words and primes?