Statistical Distributions of Prime Number Gaps

A heuristic i.e. empirical approach to the problem of prime number gaps of many kinds and types, different degrees and orders, treated as simple raw experimental data from the statistical viewpoint is presented. The aim of the article is to show a picture of the actual situation of prime number gaps in order to describe and to try to understand the structure itself of prime gaps of various kinds and orders as well as of primes themselves. The data base comprises the finite sequences of prime number gaps up to the value Pn of the prime counter n = 5∙107 that is P5E7 = P(5∙107) = 982,451,653 all of them available in the net. The statistical distributions of prime gaps are best-fitted by the pseudo-Voigt fit function, a convolution of the Lorentz and the Gauss differential distribution functions, or by the so-called E-exp or exp-exp differential distribution function or by a log-linear histogram according to the kind of gaps examined, either δiPn (higher order gaps) or ΔkPm = Pm – Pm–k (delta-lags) with i and k ≥ 2 or the simple linear differences δ1Pm = Δ1Pm = ΔPm= Pm – Pm–1 respectively. One of the unexpected results of the investigation is the appearance of inner structures at high values of nΔ, the number of the intervals of the distributions, suggesting the presence of groups or clusters strictly linked to the nature of prime numbers themselves in which the same phenomenology is present.