### Abstract:

In 1927, Emil Artin conjectured a product expression for the density of primes p for which
a given non-zero integer a is a primitive root modulo p. The conjectured density was proved
in 1967 by Hooley under the assumption of the Generalized Riemann Hypothesis. In 2014,
Lenstra, Moree, and Stevenhagen introduced a method involving character sums to deduce
the formula for the product in the density for Artin’s conjecture. The method applies in
similar problems such as the density of primes of cyclic reduction for Serre curves. In this
thesis, we introduce a generalization of this method which yields product expressions for a
large family of problems that can be stated by summations involving the orders of certain
finite groups. As a consequence, the product expressions of some Artin type problems, such
as the Titchmarsh Divisor Problem in Kummer families for primes in a given arithmetic
progression, are computed here.