Wiener-Ikehara theorem

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 $$\sum_{n=1}^\infty a(n) n^{-s}$$ where a(n) is non-negative. If the series converges to an analytic function in $$\mbox{re}(s) \ge b$$ 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 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 $$\mbox{re}(s)=1$$