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2019 Variance of arithmetic sums and $L$-functions in $\mathbb{F}_q[t]$
Chris Hall, Jonathan P. Keating, Edva Roditty-Gershon
Algebra Number Theory 13(1): 19-92 (2019). 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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Chris Hall. Jonathan P. Keating. Edva Roditty-Gershon. "Variance of arithmetic sums and $L$-functions in $\mathbb{F}_q[t]$." Algebra Number Theory 13 (1) 19 - 92, 2019. https://doi.org/10.2140/ant.2019.13.19

Information

Received: 6 April 2017; Revised: 7 August 2018; Accepted: 6 September 2018; Published: 2019
First available in Project Euclid: 27 March 2019

zbMATH: 07041706
MathSciNet: MR3917915
Digital Object Identifier: 10.2140/ant.2019.13.19

Subjects:
Primary: 11T55
Secondary: 11M38, 11M50

Rights: Copyright © 2019 Mathematical Sciences Publishers

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