Abstract
We establish asymptotic formulae for the first and second moments of quadratic Dirichlet L-functions, at the center of the critical strip, associated to the real quadratic function field \(k(\sqrt{P})\) and inert imaginary quadratic function field \(k(\sqrt{\gamma P})\) with P being a monic irreducible polynomial over a fixed finite field \(\mathbb {F}_{q}\) of odd cardinality q and \(\gamma \) a generator of \(\mathbb {F}_{q}^{\times }\). We also study mean values for the class number and for the cardinality of the second K-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over \(\mathbb {F}_{q}\). One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average L-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of L-functions over function fields.
Highlights
1 Introduction and some basic facts. It is a profound problem in analytic number theory to understand the distribution of values of L(s, χp), the Dirichlet L-functions associated to the quadratic character χp, for fixed s and variable p, where for a prime number p ≡ v with v = 1 or 3, the quadratic character χp(n) is defined by the Legendre symbol χp(n)
One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials of even degree, and in this way we are able to go beyond of what is known in the number field case
First we present some preliminary facts on quadratic Dirichlet L-functions for the rational function field k = Fq(T ) and for this we use Rosen’s book [16] as a guide to the notations and definitions
Summary
It is a profound problem in analytic number theory to understand the distribution of values of L(s, χp), the Dirichlet L-functions associated to the quadratic character χp, for fixed s and variable p, where for a prime number p ≡ v (mod 4) with v = 1 or 3, the quadratic character χp(n) is defined by the Legendre symbol. The problem about the distribution of values of Dirichlet L-functions with real characters χ modulo a prime p was first studied by Elliot [9] and later some of his results were generalized by Stankus [18]. In this context, Goldfeld and Viola [10] have conjectured an asymptotic formula for. Jutila [13] was able to establish the following asymptotic formula: Theorem 1.1 (Jutila) For v = 1 or 3, we have p≤X (log p)L. where the implied constant is not effectively calculable.
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