Constants for Artin-like problems in Kummer and division fields

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 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 \(\begin{aligned} \sum _{n=1}^{\infty }\frac{g(n)}{\#G(n)} \end{aligned}\) $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }\frac{g(n)}{\#G(n)}\end{array}$$ in which g(n) is a multiplicative arithmetic function and \(\{G(n)\}\) $\{G(n)\}$ 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.