We present some variations on of the main open problems character degrees. collect methods that have proven to be very useful work these problems. These are also solve certain zeros characters, kernels and fields values characters.

There has been some interest on how the average character degree affects structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, number zeros characters group G as in table divided irreducible $. show that if < 1 then is solvable also \frac{1}{2} supersolvable. characterise abelian groups showing \frac{1}{3} only abelian.

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, an...

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.

In this paper, under a weakened version of Hardy—Littlewood Conjecture on the number representations in Goldbach problem, we shall prove bounds for Siegel zeros real primitive Dirichlet characters composite moduli.

We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup Γ0(4) as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results....