Wiener-Ikehara theorem: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(starting exactly with my own wording from WP, will add rest in next revision)
 
imported>Richard Pinch
m (subpages)
Line 1: Line 1:
{{subpages}}
It was proved by [[Norbert Wiener]] and his student [[Shikao Ikehara]] in 1932.  It is an example of a [[Tauberian theorem]].
It was proved by [[Norbert Wiener]] and his student [[Shikao Ikehara]] in 1932.  It is an example of a [[Tauberian theorem]].



Revision as of 15:09, 9 November 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

It was proved by Norbert Wiener and his student Shikao Ikehara in 1932. It is an example of a Tauberian theorem.

Application

An important number-theoretic application of the theorem is to Dirichlet series of the form where a(n) is non-negative. If the series converges to an analytic function in with a simple pole of residue c at s=b, then .

Applying this to the logarithmic derivative of the Riemann zeta function, where the coefficints in the Dirichlet series are values of the von Mangoldt function, it is possible to deduce the prime number theorem from the fact that the zeta function has no zeroes on the line

  • S. Ikehara (1931). "An extension of Landau's theorem in the analytic theory of numbers". J. Math. Phys. 10: 1–12.