The link is between numbers and words. Both can be decomposed into primitive components: primes and lyndon words (combinatorics). The Zipf distribution takes this shape when the vocabulary is infinite:
P(r)=r^-s / Zeta(s) with s > 1
Replace r^-s with (r + a)^-s, i.e. move onto Mandelbrot's rank-shifted model of Zipf's law and Riemann's Zeta function becomes Hurwitz Zeta function (ZetaHurwitz(s,a+1) with the ranks starting at 1).
Zipf law != Zipf distribution, still this is intriguing.
> In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.
The same holds for Lyndon words decomposition of random strings to a certain extent.
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
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).
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.
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?
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.
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[ 3.1 ms ] story [ 17.2 ms ] threadZipf law != Zipf distribution, still this is intriguing.
Integers and Prime Numbers: Deriving Zipf's Law (https://arxiv.org/abs/2403.12773)
> In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.
The same holds for Lyndon words decomposition of random strings to a certain extent.
This didn't seem especially controversial of a comment, and even here there's no end of the rotten attitudes for no good reason
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
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).
> 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?