Daniele Lattanzi

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 21, 2015, Number 1, Pages 18—30

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## Details

### 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

*Χ*

^{2}

_{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*} and truncated progressions {

_{m}*n*} with

^{α}*α*∈ (1, 2) so that an injective map can be set between them through the parameter

*k*of their common fit function

*Χ*

^{2}

_{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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## Cite this paper

Lattanzi, D. (2015). Distribution of prime numbers by the modified chi-square function. Notes on Number Theory and Discrete Mathematics, 21(1), 18-30.