View source for Wiener–Ikehara theorem

{{MSC|11M45}}

A  [[Tauberian theorem]] relating the behaviour of a real sequence to the  analytic properties of the associated [[Dirichlet series]].  It is used  in the study of [[arithmetic function]]s and yields a proof of the  [[Prime number theorem]].  It was proved by [[Norbert Wiener]] and his  student [[Shikao Ikehara]] in 1932.


Let  $F(x)$ be a non-negative, [[monotone function|monotonic]] [[decreasing function]] of the positive [[real number|real]] variable $x$.  Suppose  that the [[Laplace transform]]
$$
\int_0^\infty F(x)\exp(-xs) dx
$$
converges  for $\Re s >1$ to the function $f(s)$ and that $f(s)$ is [[analytic  function|analytic]] for $\Re s \ge 1$, except for a simple [[pole]] at  $s=1$ with residue 1. Then the [[Limit of a function|limit]] as $x$ goes  to infinity of $e^{-x} F(x)$ is equal to 1.

An  important number-theoretic application of the theorem is to [[Dirichlet series]] of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is  non-negative.  If the series converges to an analytic function in $\Re s  \ge b$ with a simple pole of residue $c$ at $s = b$, then $\sum_{n\le  X}a(n) \sim c \cdot X^b$.

Applying this to the  logarithmic derivative of the [[Riemann zeta function]], where the  coefficients 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 $\Re (s)=1$.

==References==
* S. Ikehara; ''An extension of Landau's theorem in the analytic theory of numbers'',  J. Math. Phys.  '''10'''  (1931), pp. 1–12  
* N. Wiener; ''Tauberian theorems'',  Annals of Mathematics  '''33'''  (1932), pp. 1–100  
* Hugh L. Montgomery; Robert C. Vaughan; ''Multiplicative number theory I. Classical theory'',  ser. Cambridge tracts in advanced mathematics '''97'''  (2007), pp. 259–266 {{ISBN|0-521-84903-9}}