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We recall the relation between Zeta function representation of groups andtwo-dimensional topological Yang-Mills theory through Mednikh formula. We provevarious generalisations of Mednikh formulas and define generalization of Zetafunctions representations of groups. We compute some of these functions in thecase of the finite group $GL(2, {\mathbb F}_q)$ and $PGL(2,{\mathbb F}_q).$ Werecall the table characters of these groups for any $q$, compute theFrobenius-Schur indicator of their irreducible representations and give theexplicit structure of their fusion rings

Generalized Zeta function representation of groups and 2-dimensional Topological Yang-Mills theory: The example of GL(2, F_q) and PGL(2, F_q)