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We utilize harmonic analytic tools to count the number of elements of theGalois cohomology group $f\in H^1(K,T)$ with discriminant-like invariant ${\rminv}(f)\le X$ as $X\to\infty$. Specifically, Poisson summation produces acanonical decomposition for the corresponding generating series as a sum ofEuler products for a very general counting problem. This type of decompositionis exactly what is needed to compute asymptotic growth rates using a Tauberiantheorem. These new techniques allow for the removal of certain obstructions toknown results and answer some outstanding questions on the generalized versionof Malle's conjecture for the first Galois cohomology group.

Harmonic analysis and statistics of the first Galois cohomology group