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 (18)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• Low-lying zeros of families of elliptic curves
• A Symplectic Test of the L-Functions Ratios Conjecture
• Nuclei, Primes and the Random Matrix Connection
• The effect of convolving families of L-functions on the underlying group symmetries
• Low lying zeros of L-functions with orthogonal symmetry
• $n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions
• The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions
• Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves
• An Orthogonal Test of the L-Functions Ratios Conjecture
• An Orthogonal Test of the $L$-functions Ratios Conjecture, II
• A Hybrid Euler-Hadamard product formula for the Riemann zeta function
• Low-lying zeroes of Maass form $L$-functions
• A unitary test of the Ratios Conjecture
• Residue classes containing an unexpected number of primes
• The mean-square of Dirichlet L-functions
• Lower order terms in the 1-level density for families of holomorphic cuspidal newforms
• The influence of the first term of an arithmetic progression

Cited by in corpus (11)

• Low-lying zeroes of Maass form $L$-functions
• Residue classes containing an unexpected number of primes
• 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
• A disproof of Hooley's conjecture
• Determining Optimal Test Functions for Bounding the Average Rank in Families of $L$-Functions
• 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)