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 $\left(-\frac{3}{2},\frac{3}{2}\right)$, 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 $Q^{-1∕2+ϵ}$ 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 $\left[-1,1\right]$. 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 $Q^{-1∕2+ϵ}$.