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Moments of the Hurwitz zeta function on the critical line
Published online by Cambridge University Press: 28 November 2022
Abstract
We study the moments
$M_k(T;\,\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$
of the Hurwitz zeta function
$\zeta(s,\alpha)$
on the critical line,
$s = 1/2 + it$
with a rational shift
$\alpha \in \mathbb{Q}$
. We conjecture, in analogy with the Riemann zeta function, that
$M_k(T;\,\alpha) \sim c_k(\alpha) T (\!\log T)^{k^2}$
. Using heuristics from analytic number theory and random matrix theory, we conjecturally compute
$c_k(\alpha)$
. In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases
$k = 1,2$
.
MSC classification
Secondary:
11M06: $zeta (s)$ and $L(s, chi)$
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 631 - 661
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society
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