Dirichlet Series

A series with n-th term a_n n^-s, with variable s. The set of complex s for which the series converges is a half-plane to the right of the abscissa of convergence (compare with conventional power series and the radius of convergence). If the coefficients a_n are bounded by a power of n, say n^k then the series will converge to the right of the line re(s)=k+1.
A formal Dirichlet series is one considered without regard for convergence.


If a_n is multiplicative as a function of n, that is, a_mn = a_m a_n when m,n are coprime, then the series has a formal Euler product.
The series defined by such coefficients are usually referred to as L-functions.

Perhaps the most well-known Dirichlet series is the zeta-function where a_n = 1 for all n. The abscissa of convergence is 1: regarded as a function of s there is a pole at s=1.

Dirichlet series relate the behaviour of the function a_n to that of the corresponding series. For example, if A(x) is the sum of the a_n for n up to x, and F(s) is the Dirichlet series with coefficients a_n, then the rate of growth of A as a power of x is determined by the abscissa of convergence of F.

These Tauberian theorems can be viewed as relating two different weighted sums of the a_n: the weight 1/x for n up to x and zero beyond; and the weight n^-s/F(s)