Distribution of prime numbers by the modified chi-square function

Daniele Lattanzi
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 1, Pages 18–30
Full paper (PDF, 246 Kb)

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Authors and affiliations

Daniele Lattanzi
Former ENEA-FUS, Frascati, Roma, Italy
Private address: Via La Spezia 81 – 00182 Roma, Italy

Abstract

The statistical distribution of prime numbers represents an open problem in number theory still nowadays. The methodology of experimental mathematics has not yet been attempted in this field, thus the present report treats prime numbers as raw experimental data and as elements of larger and larger finite sequences {P_m}. The modified chi-square function Χ²_k(A, x/μ) with the ad-hoc A, k and μ = μ(k) parameters is the best-fit function of the differential distribution functions of both prime finite sequences {P_m} and truncated progressions {n^α} with α ∈ (1, 2) so that an injective map can be set between them through the parameter k of their common fit function Χ²_k(A, x/μ) showing that the property of scale invariance does not hold for prime distribution. The histograms of prime gaps, which are best fitted by standard statistical distribution functions, show unexpected clustering effects.

Keywords

• Prime distribution
• Prime sequences
• Modified chi-square function
• Progressions

AMS Classification

• 11N13
• 11N05
• 11B25

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