Scale Laws of Prime Number Frequencies by the Modified Chi-square Function

Abstract

The methodology of experimental mathematics has not yet been applied to the frequencies of prime numbers thus the present report treats them as raw experimental data and as elements of larger and larger finite sequences {fm}≡{mp/Pm}. The modified chi-square function Xk2(A,x/μ) with its three parameters A, k and μ=μ(k) is the best-fit function of the differential distribution functions of the finite sequences {fm} thus showing that the property of scale invariance does not hold for the statistical distributions of prime frequencies. Moreover the function Xk2(A,x/μ) with the ad-hoc values of its parameters is the best-fit function of the finite sequences of prime frequencies {fm} from the analytical viewpoint too, what leads to induction algorithms and to relationships of the kind fm~f(mp), though within the precisions of the calculations and holding locally, showing that the property of scale invariance does not hold for prime frequencies even in the analytical case. An injective map can be set between these {fm} sequences and the {nα} truncated progressions through the parameter k of their common fit function Xk2(A,x/μ) in both the statistical and the analytical case. Moreover a general experimental elementary account of Riemann’s hypothesis is given in this frame.