  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.
