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