In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in Fq[t]. The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions dr(f ) with an error term that improves the one given by Burr.

In this paper we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions L(s, χP ) where the character χ is defined by the Legendre symbol for polynomials in Fq[T ] with Fq a finite field of odd cardinality and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application we also compute the formula for the one-level density for the zeros of these L-functions.

In this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions $$L(s,\chi _{P})$$ L ( s , χ P ) where the character $$\chi $$ χ is defined by the Legendre symbol for polynomials in $$\mathbb {F}_{q}[T]$$ F q [ T ] with $$\mathbb {F}_{q}$$ F q a finite field of odd cardinality, and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application, we also compute the formula for the one-level density for the zeros of these L-functions.

We investigate the moment and the distribution of L(1, χP ), where χP varies over quadratic characters associated to irreducible polynomials P of degree 2g + 1 over Fq[T ] as g → ∞. In the first part of the paper we compute the integral moments of the class number hP associated to quadratic function fields with prime discriminants P and this is done by adapting to the function field setting some of the previous results carried out by Nagoshi in the number field setting. In the second part of the paper we compute the complex moments of of L(1, χP ) in large uniform range and investigate the statistical distribution of the class numbers by introducing a certain random Euler product. The second part of the paper is based on recent results carried out by Lumley when dealing with square-free polynomials.

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