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Moments of the Dedekind zeta function and other non-primitive L-functions

Part of: Algebraic number theory: global fields Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  15 November 2019

WINSTON HEAP
WINSTON HEAP*
Affiliation:
Department of Mathematics, University of York, York, YO10 5DD, U.K. e-mail: winstonheap@gmail.com
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Abstract

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg’s conjectures.

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11R42: Zeta functions and $L$-functions of number fields 11M50: Relations with random matrices
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 191 - 219
Copyright
© Cambridge Philosophical Society 2019

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