Abstract
Let $q$ be a positive integer ($\geq 2$), $\chi$ be a Dirichlet character modulo $q$, $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^{\prime} (s, \chi)$ denote its derivative with respect to the complex variable $s$. Let $t_{0}$ be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $|(L^{\prime}/L)(1+it_0, \chi)|$ when $\chi$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_{0} = 0$).
Funding Statement
The first author is supported by “JSPS KAKENHI Grant Number: JP25287002”. The second author is supported by the Austrian Science Fund (FWF): Projects F5507-N26, and F5505-N26 which are parts of the special Research Program “Quasi Monte Carlo Methods: Theory and Application”. Part of this work was also done while she was supported by the Japan Society for the Promotion of Science (JSPS) “Overseas researcher under Postdoctoral Fellowship of JSPS”.
Citation
Kohji MATSUMOTO. Sumaia SAAD EDDIN. "An asymptotic formula for the $2k$-th power mean value of $|(L^{\prime}/L)(1+it_0, \chi)|$." J. Math. Soc. Japan 73 (3) 781 - 814, July, 2021. https://doi.org/10.2969/jmsj/79987998
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