Dirichlet $L$-functions and primitive characters

Lecture # 2: Dirichlet L-functions, Dirichlet Characters and primes in arithmetic progressions

Dirichlet considered a question very similar to the one which inspired Euler’s introduction of the ζfunction: namely, how the primes are distributed modulo m. The simplest question of this type is whether are there infinitely many primes congruent to a modulo m. Obviously there can only be infinitely many primes of this form if a and m are relatively prime. Unfortunately, this problem turned ou...

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.

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...