Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions

Abstract

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions in the family of all characters modulo $q$, with $Q/2<q\le Q$. For test functions whose Fourier transform is supported in $(-3/2,3/2)$, we calculate this quantity beyond the square-root cancellation expansion arising from the $L$-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term $Q^{-\frac{1}{2}+\epsilon }$ in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval $[-1,1]$. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's prediction holds for any finite support up to an error $Q^{-\frac{1}{2}+\epsilon }$.

Version 1.2, 30 pages

References in corpus (4)

• $n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions
• Low-lying zeroes of Maass form $L$-functions
• The mean-square of Dirichlet L-functions
• The influence of the first term of an arithmetic progression

Cited by in corpus (10)

• Low-lying zeroes of Maass form $L$-functions
• Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture
• Low-lying zeros of quadratic Dirichlet $L$-functions: Lower order terms for extended support
• One-level density estimates for Dirichlet L-functions with extended support
• Determining optimal test functions for $2$-level densities
• Determining Optimal Test Functions for Bounding the Average Rank in Families of $L$-Functions
• A disproof of Hooley's conjecture
• Fredholm Theory and Optimal Test Functions for Detecting Central Point Vanishing Over Families of L-functions
• On the moments of one-level densities in families of holomorphic cusp forms in the level aspect
• Explicit formulas for the spectral side of the trace formula of SL(2)