One-level density estimates for Dirichlet L-functions with extended support (2002.11968v2)

Abstract: We estimate the $1$-level density of low-lying zeros of $L(s,\chi)$ with $\chi$ ranging over primitive Dirichlet characters of conductor $\in [Q/2,Q]$ and for test functions whose Fourier transform is supported in $[- 2 - 50/1093, 2 + 50/1093]$. Previously any extension of the support past the range $[-2,2]$ was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the Generalized Riemann Hypothesis (e.g Montgomery's conjecture). Our work provides the first example of a family of $L$-functions in which the support is unconditionally extended past the "trivial range" that follows from a simple application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for non-vanishing of $L(s,\chi)$.