The integral moments and ratios of quadratic Dirichlet L-functions over monic irreducible polynomials in $$\mathbb {F}_{q}[T]$$

Abstract

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.

Moments of quadratic Dirichlet $L$-functions over Function Fields

“…Extending the recipe developed by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith [14] to the function field case, J. C. Andrade, H. Jung and A. Shamesaldeen [5] conjectured for every integer k ≥ 1, there exists an polynomial Q k , with an explicit expression, of degree k(k + 1)/2 such that, as g → ∞,…”

Moments of the Hurwitz zeta function on the critical line

“…Such a hybrid Euler–Hadamard product was proved by Bui and Keating [ 9 ] in their study of moments in the q -aspect of Dirichlet L -functions at the central point (see [ 9 , remark 1]). Similar hybrid Euler–Hadamard products have been used in the literature for studying moments in many other contexts such as for for orthogonal and symplectic families of L -functions [ 10 ]; for [ 8 ]; for the Dedekind zeta function of a Galois extension of [ 23 ]; for quadratic Dirichlet L -functions over function fields [ 7 ], [ 3 ]; for normalised symmetric square L -functions associated with eigenforms [ 16 ]; and for quadratic Dirichlet L -functions over function fields associated to irreducible polynomials [ 2 ].…”