Mean value theorems for $L$-functions over prime polynomials for the rational function field
Volume 161 / 2013
Acta Arithmetica 161 (2013), 371-385
MSC: Primary 11G20; Secondary 11M38, 11M50, 14G10.
DOI: 10.4064/aa161-4-4
Abstract
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.
