Set of Wiener-Ikehara

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The Wiener-Ikehara theorem (sometimes also called Taubätze von Wiener-Ikehara ) is a mathematical theorem, which is particularly used in analytical number theory . Under certain conditions he makes statements about the asymptotic behavior of number theoretic functions. It is named after Norbert Wiener and Shikao Ikehara .

statement

Let it be given on the half plane by the Dirichlet series

being for everyone . I also have the function

for a continuous continuation on the closed half-plane . Then already applies

.

Version for integrals

Let it be a real-valued function that fulfills the following properties:

  • it is increasing monotonously,
  • it disappears for all values ,
  • it is steady to the right.

The Mellin-Stieltjes transform continues to exist

for all values . There is now one so that the function

allows to continue steadily on the half-level , it already applies

.

example

A simple example is provided by the Riemann zeta function , which is on the half plane through the standard Dirichlet series

given is. It can be continued to a holomorphic function and has a first order pole with a residual . It follows that

is a whole function, so in particular it can be continued from continuously to the half-plane . Indeed

application

With the help of Wiener-Ikehara's pigeon theorem , the prime number theorem can be proven. The theorem is applied to the Dirichlet series of the function , whereby it must first be shown that the zeta function does not vanish on the straight line . It follows

which is equivalent to the prime number theorem.

Generalizations

In 1954 Delange was able to generalize the Wiener-Ikehara theorem clearly, namely to singularities of mixed type. Let it be a Dirichlet series with non-negative coefficients which converges on a half plane . Assume that , with the exception of the point , it can be continued holomorphically to the entire straight line and that in a small area it is in the form

lets write, being a real number and the functions and are holomorphic with . Then: if there is no negative integer, it follows

and is it a negative integer and :

literature

  • Jacob Korevaar: Tauberian Theory. A century of developments. Basic teaching of the mathematical sciences, Springer-Verlag, Berlin Heidelberg New York, ISBN 3-540-21058-X .
  • S. Ikehara: An extension of Landau's theorem in the analytic theory of numbers , Journal of Mathematics and Physics of the Massachusetts Institute of Technology, Volume 10, 1931, pp. 1-12
  • Norbert Wiener: Tauberian Theorems , Annals of Mathematics, Second Series, Volume 33, 1932, pp. 1-100

Individual evidence

  1. ^ Gérald Tenenbaum: Introduction to analytic and probabilistic number theory , AMS, 1990, p. 350