Density Functions for Families of Dirichlet Characters

As M. Rubinstein [Ru] has already considered all primitive quadratic characters with prime conductor q ∈ [N, 2N ], we do not include our notes of this case. For these families we show the 1-Level Densities agree with the Unitary Group for even Schwartz functions φ̂ with supp(φ̂) ⊂ (−2, 2). We conclude with an appendix on “reasonable” assumptions which would allow us to extend the support of the test functions, possibly up to (−4, 4). These notes were written between 1999 and 2000 as a precursor to my dissertation; they are meant to be a rapid introduction to the calculations, with little motivation. For more on such calculations, see the paper by C. Hughes and Z. Rudnick [HR], where the first family was independently considered (and by additional arguments they obtained results for supp(φ̂) ⊂ [−2, 2]). For motivation as to why one considers such quantities, see [ILS, Mil, Ru].