Vol. 13, No. 1, 2019

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Variance of arithmetic sums and $L$-functions in $\mathbb{F}_q[t]$

Chris Hall, Jonathan P. Keating and Edva Roditty-Gershon

Vol. 13 (2019), No. 1, 19–92
DOI: 10.2140/ant.2019.13.19
Abstract

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain L-functions of degree 2 and higher in Fq[t], in the limit as q . This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 L-functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over Fq[t].

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Keywords
$L$-functions, Mellin transform
Mathematical Subject Classification 2010
Primary: 11T55
Secondary: 11M38, 11M50
Milestones
Received: 6 April 2017
Revised: 7 August 2018
Accepted: 6 September 2018
Published: 13 February 2019
Authors
Chris Hall
Department of Mathematics
University of Western Ontario
London, ON
Canada
Jonathan P. Keating
School of Mathematics
University of Bristol
Bristol
United Kingdom
Edva Roditty-Gershon
Department of Applied Mathematics
Holon Institute of Technology
Holon
Israel