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

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

#21

Post by linksnapzz »

smokedetector1 wrote: post #16
IAmBroom wrote: post #13

Earlier quoted context omitted.

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 Ta…

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?

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

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

#22

Post by openasocket »

briandw wrote: post #15

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

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

#23

Post by charlieyu1 »

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.

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

#24

Post by cwmoore »

smokedetector1 wrote: post #16
IAmBroom wrote: post #13

Earlier quoted context omitted.

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 Ta…

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?

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

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

#25

Post by daoboy »

daoboy wrote: post #8

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

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

#26

Post by openasocket »

lioeters wrote: post #12
Sharlin wrote: post #10

Earlier quoted context omitted.

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

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 La…

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.

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

#27

Post by jesuslop »

dist-epoch wrote: post #17
briandw wrote: post #15

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.

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 an…

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

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

#28

Post by adgjlsfhk1 »

seanhunter wrote: post #14
zombot wrote: post #4

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_…

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

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

#29

Post by lioeters »

openasocket wrote: post #26
lioeters wrote: post #12

Earlier quoted context omitted.

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 La…

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.

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

#30

Post by smokedetector1 »

cwmoore wrote: post #24
smokedetector1 wrote: post #16

Earlier quoted context omitted.

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?

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?
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