The integral moments and ratios of quadratic Dirichlet \(L\)-functions over monic irreducible polynomials in \(\mathbb{F}_q [T]\). (English) Zbl 1485.11137
Summary: 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 [J. B. Conrey et al., Proc. Lond. Math. Soc. (3) 91, No. 1, 33–104 (2005; Zbl 1075.11058)]. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer [B. Conrey et al., Commun. Number Theory Phys. 2, No. 3, 593–636 (2008; Zbl 1178.11056)] 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 \(L(s,\chi_P)\) where the character \(\chi\) is defined by the Legendre symbol for polynomials in \(\mathbb{F}_q [T]\) with \(\mathbb{F}_q\) 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.
MSC:
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11G20 | Curves over finite and local fields |
| 11M50 | Relations with random matrices |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
Keywords:
function fields; integral moments of \(L\)-functions; quadratic Dirichlet \(L\)-functions; ratios conjectureReferences:
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