Mean value theorems for $L$-functions over prime polynomials for the rational function field

Julio C. Andrade ¹   ; Jonathan P. Keating ²

¹ Institute for Computational and Experimental Research in Mathematics (ICERM) Brown University 121 South Main Street Providence, RI 02903, U.S.A.
² School of Mathematics University of Bristol Bristol BS8 1TW, UK

Acta Arithmetica, Tome 161 (2013) no. 4, pp. 371-385

Résumé

The first and second moments are established for the family of quadratic Dirichlet $L$-functions over the rational function field at the central point ${s=1/2}$, where the character $\chi $ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomials $P$ of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of $P$ is large. The first moment obtained here is the function field analogue of a result due to Jutila in the number field setting. The approach is based on classical analytical methods and relies on the use of the analogue of the approximate functional equation for these $L$-functions.

Détail

DOI : 10.4064/aa161-4-4

Keywords: first second moments established family quadratic dirichlet l functions rational function field central point where character chi defined legendre symbol polynomials finite fields runs monic irreducible polynomials given odd degree asymptotic formulae derived fixed finite fields degree large first moment obtained here function field analogue result due jutila number field setting approach based classical analytical methods relies analogue approximate functional equation these l functions