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

Julio C. Andrade, Jonathan P. Keating (2013)

Acta Arithmetica

The first and second moments are established for the family of quadratic Dirichlet L-functions over the rational function field at the central point s=1/2, where the character χ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomials P of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of P is large. The first moment obtained here is the function field analogue of a result due to Jutila in the...

On the orthogonal symmetry of L-functions of a family of Hecke Grössencharacters

J. B. Conrey, N. C. Snaith (2013)

Acta Arithmetica

The family of symmetric powers of an L-function associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and p-adic points of view. Here we examine one explicit such family from the perspectives of classical analytic number theory and random matrix theory, especially focusing on evidence for the symmetry type of the family. In particular, we investigate the values at the central point and give evidence that this family can be modeled by...

The n-level densities of low-lying zeros of quadratic Dirichlet L-functions

Jake Levinson, Steven J. Miller (2013)

Acta Arithmetica

Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions f̂₁, ..., f̂ₙ supported in i = 1 n | u i | < 2 , and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form,...

The second moment of quadratic twists of modular L-functions

K. Soundararajan, Matthew P. Young (2010)

Journal of the European Mathematical Society

We study the second moment of the central values of quadratic twists of a modular L -function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.

The shifted fourth moment of automorphic L-functions of prime power level

Olga Balkanova (2016)

Acta Arithmetica

We prove an asymptotic formula for the fourth moment of automorphic L-functions of level p ν , where p is a fixed prime number and ν → ∞. This is a continuation of work by Rouymi, who computed the asymptotics of the first three moments at a prime power level, and a generalization of results obtained for a prime level by Duke, Friedlander Iwaniec and Kowalski, Michel VanderKam.

Triple correlation of the Riemann zeros

J. Brian Conrey, Nina C. Snaith (2008)

Journal de Théorie des Nombres de Bordeaux

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously...

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