# Dirichlet Generating function and Poles.

1. Jul 13, 2007

### Klaus_Hoffmann

Let be:

$$(1/T) \int_{-T}^{T}dtF(s+it)G(s-it)= \sum_{1 \le n} a(n)b(n)n^{-2s}$$
valid for s >1.

My question is if at s=1 $$\sum_{1 \le n} a(n)b(n)n^{-2s}$$ has a pole, then my question is if then the integral

$$(1/T) \int_{-T}^{T}dtF(0.5+it)G(0.5-it)$$ will be divergent.

where F(s) and G(s) are the Dirichlet generating functions for a(n) and b(n)

so if we have the Dirichlet series with coefficients a(n)b(n) and we want to know if $$a(n)b(n)n^{-1}$$ is divergent we could perform the integral

$$(1/T) \int_{-T}^{T}dtF(0.5+it)G(0.5-it)$$ as T---> infinity

to see if it diverges