# A Symplectic Test of the L-Functions Ratios Conjecture

## TL;DR

This paper rigorously tests the L-functions Ratios Conjecture for the 1-level density of quadratic Dirichlet L-functions, confirming its predictions for lower order terms in a specific symplectic family.

## Contribution

It provides the first detailed verification of the Ratios Conjecture's lower order terms for the 1-level density in a symplectic family of L-functions.

## Key findings

- Confirmed the conjecture's predictions for lower order terms up to O(X^{-1/2+epsilon})
- Achieved perfect agreement with the Ratios Conjecture for test functions supported in (-1/3, 1/3)
- Validated the conjecture up to errors of size O(X^{-epsilon}) for test functions supported in (-1, 1)

## 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.