A Symplectic Test of the L-Functions Ratios Conjecture

Abstract

Recently Conrey, Farmer and Zirnbauer conjectured formulas for the averages over a family of ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from n-level correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The L-functions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1-level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d \le X. For test functions supported in (-1/3, 1/3) we calculate all the lower order terms up to size O(X^{-1/2+epsilon}) and observe perfect agreement with the conjecture (for test functions supported in (-1, 1) we show agreement up to errors of size O(X^{-epsilon}) for any epsilon). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture's prediction for the 1-level density.

29 pages, version 1.3 (corrected a typo in the proof of Lemma 3.2 and a few other typos, updated some references). To appear in IMRN

References in corpus (3)

• The effect of convolving families of L-functions on the underlying group symmetries
• $n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions
• On the spacing distribution of the Riemann zeros: corrections to the asymptotic result

Cited by in corpus (25)

• Nuclei, Primes and the Random Matrix Connection
• The effect of convolving families of L-functions on the underlying group symmetries
• An Orthogonal Test of the L-Functions Ratios Conjecture
• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
• An Orthogonal Test of the $L$-functions Ratios Conjecture, II
• A unitary test of the Ratios Conjecture
• One level density of low-lying zeros of quadratic and quartic Hecke $L$-functions
• One level density of low-lying zeros of families of $L$-functions
• Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-Functions
• Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture
• Autocorrelation of ratios of L-functions
• Determining optimal test functions for $2$-level densities
• Modeling Convolutions of $L$-Functions
• The quadratic character experiment
• Traces of high powers of the Frobenius class in the hyperelliptic ensemble
• One level density of low-lying zeros of quadratic Hecke $L$-functions of Imaginary Quadratic Number Fields
• One-level density of quadratic twists of $L$-functions
• One-level density and non-vanishing for cubic $L$-functions over the Eisenstein field
• Lower order terms of the one level density of a family of quadratic Hecke $L$-functions
• Low-lying Zeros of Number Field $L$-functions
• One level density of low-lying zeros of quadratic Hecke $L$-functions to prime moduli
• Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters
• Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory
• On the Distribution of Zeroes of Artin-Schreier L-functions
• An elliptic curve test of the L-Functions Ratios Conjecture