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http://dx.doi.org/10.7858/eamj.2019.009

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

Jo, Sihun (Department of Mathematics Education, Woosuk University)
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Abstract
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.
Keywords
Dirichlet L-function; holomorphic cusp form; Poisson distribution;
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Times Cited By KSCI : 1  (Citation Analysis)
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