Random Characters under the $L$-Measure, I: Dirichlet Characters

A Note on Dirichlet Characters

Denoting by r(k, m, p) the first occurrence of m consecutive fcth power residues of a prime p = 1 (mod k), we show that r(k, m, p) > c log p for infinitely many p (c is an absolute constant) provided that k is even and m ä 3.

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...

Math 254a: the Analytic Class Number Formula, and Dirichlet Characters

Heritability of Threshold Characters*i

H E amount of selection pressure applied on the phenotypic level to T metric characters can be expressed in terms of the selection differential, defined as the extent of superiority of the individuals which are the parents of the next generation (weighted for the relative number of offspring they produce) over the average of their own generation. For characters showing a discontinuous distribut...