Julio Andrade

The mean value of L(12,χ) in the hyperelliptic ensemble

Abstract We obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field at the central point s = 1 2 . Specifically, we compute the expected value of L ( 1 2 , χ ) for an ensemble of hyperelliptic curves of genus g over a fixed finite field as g → ∞ . Our approach relies on the use of the analogue of the approximate functional equation for such L-functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of L-functions motivated by Random Matrix Theory.

Conjectures for the integral moments and ratios of L-functions over function fields

Abstract We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of L-functions defined over number fields. Specifically, we give a heuristic for the moments and ratios of a family of L-functions associated with hyperelliptic curves of genus g over a fixed finite field F q in the limit as g → ∞ . Like in the number field case, there is a striking resemblance to the corresponding formulae for the characteristic polynomials of random matrices. As an application, we calculate the one-level density for the zeros of these L-functions.

A NOTE ON THE MEAN VALUE OF L-FUNCTIONS IN FUNCTION FIELDS

An asymptotic formula for the sum

Newman's conjecture in various settings ☆

\sum L(1,\chi)

Rudnick and Soundararajan's theorem for function fields

is established for a family of hyperelliptic curves of genus

HILBERT–PÓLYA CONJECTURE, ZETA FUNCTIONS AND BOSONIC QUANTUM FIELD THEORIES

g

Mean value theorems for L-functions over prime polynomials for the rational function field

over a fixed finite field

Special Sets of Primes in Function Fields

\mathbb{F}_q

Truncated Product Representations for L-Functions in the Hyperelliptic Ensemble

as

Average values of L-series for real characters in function fields

g\rightarrow\infty