  If the prime numbers are pseudo-randomly distributed, then analogy with
quantum systems suggests that counting primes might be modeled by a
non-homogeneous Poisson process. Consequently, postulating underlying gamma
statistics, more-or-less standard heuristic arguments borrowed from quantum
mechanics in the context of functional integration allows to derive analytic
expressions of several average counting functions associated with prime
numbers. The expressions are certain sums of incomplete gamma functions that
are closely related to logarithmic-type integral functions --- which in turn
are well-known to give the asymptotic dependence of the various counting
functions up to error terms. The relatively broad success of quantum heuristics
applied to functional integrals in general along with the excellent numerical
accuracy of the analytic expressions for the average counting functions provide
strong evidence of a gamma distribution for prime powers.
