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I have been reading about the applications of the Riemann zeta function in physics and came across something called a "moment". I have never heard of such a property of the Riemann zeta function so I tried to find information on it on the Internet, without success. Neither the Wikipedia page nor other articles define what a $\zeta$ "moment" is.

For instance, on the website of the University of Bristol, there is the following text:

Now, there are certain attributes of the Riemann zeta function called its moments which should give rise to a sequence of numbers.

One could not be more vague. Further on, one can read:

(...) only two of these moments were known: 1, as calculated by Hardy and Littlewood in 1918; and 2, calculated by Ingham in 1926.

I was unable to find references to these "calculations" by H&L and Ingham. Even more puzzling is:

The next number in the series was suggested as 42 by Conrey (...). The challenge for the quantum physicists then, was to use their quantum methods to check the number 42.

This makes no sense at all. What "quantum methods" are we talking about and what does "check the number 42" mean? I understand they didn't want to go into too much detail but this is suitably vague to confuse any reader.

So what is a "moment" of the Riemann zeta function and why is it important?

Klangen
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    Just a thought... The moments of a function has a specific definition, so presumably you can just appliy it, e.g. see https://en.wikipedia.org/wiki/Moment_(mathematics) then inter its meaning from knowledge of the function and the definition of moment. – pshmath0 Jul 21 '18 at 16:21
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    I have the exact same question, not much of an answer out there yet. – Ponce Nov 12 '20 at 20:30

1 Answers1

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Raziwill has a paper about the moments of the Riemann Zeta function:

The 4.36-th moment of the Riemann Zeta function

Both Ingham's paper of 1926 and Hardy-Littlewood's paper of 1918 are in the references.

Radom matrix theory (and quantum billiard) is known to be related to the spacings between the non-trivial roots of the Riemann zeta function:

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

Edit: To address the question why the moments of the Riemann zeta function are important - this is really involved with a lot of analytic number theory and the Riemann hypothesis. In fact, the quality of the asymptotic formula depends on RH. Since the latter is one of the most important conjectures in number theory, also the moments and other properties of the zeta function are important. The exact connections are a bit technical. The introduction of the paper Moments of the Riemann zeta function by Soundararajan give a good survey on this. There is also a connection to random matrix theory, i.e., to the link above. This makes it important, too.

Dietrich Burde
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  • Thank you. The paper you linked provides a definition of a moment in terms of an integral, but as many other papers it fails to describe what it is for, or why it is important. If you can provide these answers in your post then I will award the bounty. – Klangen Jul 16 '18 at 11:27
  • @PreservedFruit Moments of the Riemann zeta functions give insight in the behaviour of the Riemann zeta function, and its asymptotic growth is related to the Riemann hypothesis. This makes it important (assuming you find RH important, of course). I added a few lines along with a reference, which explains this a bit more precise. – Dietrich Burde Jul 21 '18 at 16:17
  • I agree, there is a surprisingly lack of explanations even in introductory papers as to why moments are interesting, in particular what kind of insight give your their values about the Riemann zeta function. – Ponce Nov 12 '20 at 20:29