One-level density estimates for Dirichlet L-functions with extended support

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 t...

Calculation of Dirichlet L-Functions

A method for calculating Dirichlet L-series is presented along with the theory of residue class characters and their automatic generation. Tables are given of zeros of Lseries for moduli S 24.

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-lev...

On the Order of Dirichlet L-functions

Real zeros of quadratic Dirichlet L - functions

A small part of the Generalized Riemann Hypothesis asserts that L-functions do not have zeros on the line segment ( 2 , 1]. The question of vanishing at s = 2 often has deep arithmetical significance, and has been investigated extensively. A persuasive view is that L-functions vanish at 2 either for trivial reasons (the sign of the functional equation being negative), or for deep arithmetical r...