Extending the Support in the 1-level Density for Families of Dirichlet Characters

We study the distribution of the zeros near the central point for following families of primitive Dirichlet characters: (1) all primitive characters of conductor m, m a fixed prime; (2) all primitive characters of conductor m, m an odd square-free number with r factors (r fixed); (3) all primitive characters whose conductor is a square-free odd integer m ∈ [N, 2N ]. For these families the 1-level densities agree with the Unitary Group for even Schwartz functions φ̂ with supp(φ̂) ⊂ (−2, 2). We investigate the consequences of conjectures about the modulus dependence in the error terms in the distribution of primes in congruence classes. We show how some natural conjectures imply the 1-level densities agree with Unitary matrices for arbitrary support. Further, we show how some weaker conjectures still give an improvement over (−2, 2), allowing support up to (−4, 4). These are very rough notes.