Dirichlet Series - Analytic Properties of Dirichlet Series: The Abscissa of Convergence

Analytic Properties of Dirichlet Series: The Abscissa of Convergence

Given a sequence {a_n}_{n ∈ N} of complex numbers we try to consider the value of

as a function of the complex variable s. In order for this to make sense, we need to consider the convergence properties of the above infinite series:

If {a_n}_{n ∈ N} is a bounded sequence of complex numbers, then the corresponding Dirichlet series f converges absolutely on the open half-plane of s such that Re(s) > 1. In general, if a_n = O(nk), the series converges absolutely in the half plane Re(s) > k + 1.

If the set of sums a_n + a_{n + 1} + ... + a_{n + k} is bounded for n and k ≥ 0, then the above infinite series converges on the open half-plane of s such that Re(s) > 0.

In both cases f is an analytic function on the corresponding open half plane.

In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex plane, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes.

In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain.