Double coset zeta function of some p-groups of maximal class

The study of zeta functions has long been a central theme in number theory and group theory, providing deep insights into the structure and properties of various algebraic objects. Starting with the subgroup zeta function, a possible question is what happens if we replace the number of cosets with the number of double cosets. This question leads to the definition of the doube coset zeta function. The aim is to study the double coset zeta function of a given group or of some (families of) finite groups. At first we will recall some known results on the subgroup zeta function, in particular we will analyse two examples: the direct product of d infinite cyclic groups, where d is a natural number, and the discrete Heisenberg group. Then, after an introduction to double cosets, we will give a way to define the double coset zeta function and we will study some examples in detail. In particular, we will analyse the dihedral groups of order powers of two, the semidihedral groups and the quaternion groups of the same order. Moreover, we will move to the pro-2-dihedral group and then we will generalise the result for the pro-p-dihedral groups with a general prime p.