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Mean values of derivatives of L-functions in function fields: III

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  27 December 2018

Julio Andrade
Julio Andrade*
Affiliation:
Department of Mathematics, University of Exeter, Exeter, EX4 4QF, UK (j.c.andrade@exeter.ac.uk)
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Abstract

In this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

Keywords

function fieldsirreducible polynomialshyperelliptic curvesderivatives of L–functionsmoments of L–functionsquadratic Dirichlet L–functionsrandom matrix theory

MSC classification

Primary: 11M38: Zeta and $L$-functions in characteristic $p$
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11G20: Curves over finite and local fields 11M50: Relations with random matrices 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 905 - 913
Copyright
Copyright © Royal Society of Edinburgh 2018 

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