Constants for Artin like problems in Kummer and division fields

TL;DR

This paper uses character sums to explicitly compute constants related to the Titchmarsh divisor problem in Kummer and division fields, extending previous methods to non-density constants.

Contribution

It introduces a general framework for evaluating constants in number theory problems using character sums, applicable to Kummer and division fields.

Findings

Explicit formulas for Titchmarsh divisor constants in specific fields

Extension of character sums method to non-density constants

General product expression for sums involving Galois groups

Abstract

We apply the character sums method of Lenstra, Moree, and Stevenhagen, to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and for division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form $$\sum_{n=1}^{\infty}\frac{g(n)}{\#G(n)}$$ in which $g(n)$ is a multiplicative arithmetic function and $\{G(n)\}$ is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants.