What does the Riemann zeta function have to do with the distribution of primes?

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mmb1699 3 days ago 30 commentsRead Article on hidden-phenomena.com

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lioeters•about 14 hours 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.

bradrn•about 16 hours ago

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...

briandw•about 12 hours ago

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•about 9 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•about 11 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)

https://www.youtube.com/watch?v=aT0VbxAUwNA

jesuslop•about 8 hours ago

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

dadoum•about 7 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?

charlieyu1•about 9 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•about 15 hours ago

Stupid question on toilet: if prime numbers are 1s and other integers are 0s. What do we get from this “digital” landscape?

Sharlin•about 14 hours ago

If you plot the positive integers two-dimensionally in a square spiral arrangement, eg.

  5 4 3
  6 1 2
  7 8 9

and so on, and mark all primes, what you get is the Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiral

lioeters•about 13 hours ago

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•about 9 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.

meindnoch•about 15 hours ago

The first-order difference of the prime-counting function: https://en.wikipedia.org/wiki/Prime-counting_function

soupspaces•about 13 hours ago

The decimal expansion of an irrational number https://oeis.org/A010051

dist-epoch•about 11 hours ago

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).

pishpash•about 10 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•about 15 hours 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•about 13 hours 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.

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...

adgjlsfhk1•about 8 hours ago

that definition only converges for Re(z)>1. for Re(z)<=1, you need alternate formulas

daoboy•about 14 hours ago

Although the author turned out to be a fairly despicable person, Prime Obsession is an absolutely wonderful book on this subject.

daoboy•about 9 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•about 7 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•about 13 hours 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•about 12 hours 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?

cwmoore•about 9 hours ago

Is it not obvious? Replace “fairly despicable” with “white nationalist” instead of whatever anyone might imagine could make a mathematician despicable?

linksnapzz•about 10 hours ago

I always read that word as though it were being pronounced by Daffy Duck; "you're dethpikable!".

Xmd5a•about 16 hours ago

What is the relationship between Riemann zeta function and Zipf's law? And between words and primes?