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# Wiener-Ikehara Theorem Tweet

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1. Wiener-Ikehara Theorem
Hi,

I'm doing an undergraduate project about the distribution of the periodic orbits of a dynamical system. There is an asymptotic result about their distribution that greatly resembles the asymptotic result about the distribution of the primes.

At one point in the project, I have to use the Wiener-Ikehara Theorem (as is done in (some proofs of) the Prime Number Theorem). It is a particular Tauberian Theorem. I've attatched it to this post (in the statement, A is a constant). The integral involved is a Stieltjes integral.

I would like to find a proof of the theorem online (or just anything that tells me more about it). I cannot find even a statement online (the one I have attatched is from a book that quotes it exactly as I've written it, and don't talk about it, they just use it when necessary but without much explanation) so that I can have a better idea of what's going on. Could anyone tell me if they know where to find that result online?

Also, can anyone tell me if the conclusion of the theorem would be true if all I knew was that I had equality only for a puntured disc about 1, but with the function phi being analytic in the whole disc?

Thanks a lot.
Attached Files
2. wiener-ikehara.pdf (63.5 KB, 42 views)
3. Re: Wiener-Ikehara Theorem
FYI. Cross posted on Nrich.

Nrich
4. Re: Wiener-Ikehara Theorem
Hi,

I'm doing an undergraduate project about the distribution of the periodic orbits of a dynamical system. There is an asymptotic result about their distribution that greatly resembles the asymptotic result about the distribution of the primes.

At one point in the project, I have to use the Wiener-Ikehara Theorem (as is done in (some proofs of) the Prime Number Theorem). It is a particular Tauberian Theorem. I've attatched it to this post (in the statement, A is a constant). The integral involved is a Stieltjes integral.

I would like to find a proof of the theorem online (or just anything that tells me more about it). I cannot find even a statement online (the one I have attatched is from a book that quotes it exactly as I've written it, and don't talk about it, they just use it when necessary but without much explanation) so that I can have a better idea of what's going on. Could anyone tell me if they know where to find that result online?

Also, can anyone tell me if the conclusion of the theorem would be true if all I knew was that I had equality only for a puntured disc about 1, but with the function phi being analytic in the whole disc?

Thanks a lot.
http://www.cecm.sfu.ca/~pborwein/pbo...ources/pnt.pdf
5. Re: Wiener-Ikehara Theorem
Thanks very much! That has helped me; it confirms what I want, cheers!