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June 2013 On the Fourth Moment of the Epstein Zeta Functions and the Related Divisor Problem
Keiju SONO
Tokyo J. Math. 36(1): 269-287 (June 2013). DOI: 10.3836/tjm/1374497524

Abstract

In this paper, we study the fourth moment of the Epstein zeta function $\zeta (s;Q)$ associated to a $n\times n$ positive definite symmetric matrix $Q$ ($n\geq 4$) on the line $\mathrm{Re}(s)=\frac{n-1}{2}$. We prove that the integral $\int _{0}^{T}|\zeta (\frac{n-1}{2}+it;Q)|^{4}dt$ is evaluated by $O(T(\mathrm{log}\,T)^{4})$ if $Q$ satisfies some conditions. As an application, we consider the divisor problem with respect to the coefficients of the Dirichlet series of Epstein zeta functions. Certain estimates for the error term of the sum of the Dirichlet coefficients are obtained by combining our results and Fomenko's estimates for $\zeta (\frac{n-1}{2}+it;Q)$.

Citation

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Keiju SONO. "On the Fourth Moment of the Epstein Zeta Functions and the Related Divisor Problem." Tokyo J. Math. 36 (1) 269 - 287, June 2013. https://doi.org/10.3836/tjm/1374497524

Information

Published: June 2013
First available in Project Euclid: 22 July 2013

zbMATH: 1355.11027
MathSciNet: MR3112388
Digital Object Identifier: 10.3836/tjm/1374497524

Rights: Copyright © 2013 Publication Committee for the Tokyo Journal of Mathematics

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Vol.36 • No. 1 • June 2013
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