A Gamma Distribution Hypothesis for Prime $k$-tuples

We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples. Possessing average counting functions along with the exact functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae.