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bwfan123 20 minutes ago [-]
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.
lioeters 1 days ago [-]
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.
openasocket 22 hours ago [-]
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
dist-epoch 24 hours ago [-]
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.
[2] Prime numbers and the Riemann hypothesis. Mazur, Stein
dadoum 19 hours ago [-]
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?
18 hours ago [-]
charlieyu1 21 hours ago [-]
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.
markus_zhang 1 days ago [-]
Stupid question on toilet: if prime numbers are 1s and other integers are 0s. What do we get from this “digital” landscape?
Sharlin 1 days ago [-]
If you plot the positive integers two-dimensionally in a square spiral arrangement, eg.
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.
openasocket 21 hours ago [-]
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.
lioeters 20 hours ago [-]
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.
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).
pishpash 22 hours ago [-]
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.
zombot 1 days ago [-]
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...
seanhunter 1 days ago [-]
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.
I'm just not mathematician enough to convert that into a C function. Let alone one that is both efficient and reasonably accurate.
daoboy 1 days ago [-]
Although the author turned out to be a fairly despicable person, Prime Obsession is an absolutely wonderful book on this subject.
daoboy 21 hours ago [-]
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
cwmoore 19 hours ago [-]
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.
IAmBroom 1 days 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
smokedetector1 1 days ago [-]
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?
IAmBroom 3 hours ago [-]
Obviously, I would prefer OC was more specific. I said as much.
smokedetector1 1 hours ago [-]
Genuine question: are you on the spectrum? I find this reaction strange
linksnapzz 22 hours ago [-]
I always read that word as though it were being pronounced by Daffy Duck; "you're dethpikable!".
cwmoore 21 hours ago [-]
Is it not obvious? Replace “fairly despicable” with “white nationalist” instead of whatever anyone might imagine could make a mathematician despicable?
smokedetector1 20 hours ago [-]
Why does this shorthand upset you?
cwillu 5 hours ago [-]
A shorthand is generally unambiguous in context; this was not.
smokedetector1 55 minutes ago [-]
Why does it matter to you so much that OC is unambiguous?
cwmoore 19 hours ago [-]
I am not upset. It is obviously obstusely, impenetrably vague. What is your problem?
Xmd5a 1 days ago [-]
What is the relationship between Riemann zeta function and Zipf's law? And between words and primes?
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 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
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
[1] https://en.wikipedia.org/wiki/Von_Mangoldt_function
[2] Prime numbers and the Riemann hypothesis. Mazur, Stein
> 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?
> 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.
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.
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).
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...
https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-...
This didn't seem especially controversial of a comment, and even here there's no end of the rotten attitudes for no good reason
Read-only or read never is looking wiser than it did a few years ago.
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