THE ASYMPTOTIC BEHAVIOUR OF THE AVERAGING VALUE OF SOME DIRICHLET SERIES USING POISSON DISTRIBUTION

THE ASYMPTOTIC BEHAVIOUR OF THE AVERAGING VALUE OF SOME DIRICHLET SERIES USING POISSON DISTRIBUTION

We investigate the averaging value of a random sampling of a Dirichlet series with some condition using Poisson distribution. Our result is the following: Let $L(s)={\sum}^{\infty}_{n=1}{\frac{a_n}{n^s}}$ be a Dirichlet series that converges absolutely for Re(s) > 1. If $X_t$ is an increasing random sampling with Poisson distribution and there exists a number $0<{\alpha}<{\frac{1}{2}}$ such that ${\sum}_{n{\leq}u}a_n{\ll}u^{\alpha}$, then we have $${\mathbb{E}}L(1/2+iX_t)=O(t^{\alpha}{\sqrt{{\log}t}})$$, for all sufficiently large t in ${\mathbb{R}}$. As a result, we get the behaviour of $L({\frac{1}{2}}+it)$ such that L is a Dirichlet L-function or a modular L-function, when t is sampled by the Poisson distribution.