1307.1717 (J. LaChapelle)

Evidence of a Gamma Distribution for Prime Numbers    [PDF]

J. LaChapelle

If the occurrence of prime numbers is a random process, then analogy with quantum systems suggests that a gamma distribution governs the primes. Consequently, postulating underlying gamma statistics in the context of functional integration, more-or-less standard heuristic arguments from quantum mechanics 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 agreement of the subsequent analytic expressions obtained for the average counting functions provide strong evidence of a gamma distribution for prime numbers.

View original: http://arxiv.org/abs/1307.1717