Fractal in the statistics of Goldbach partition

TL;DR

This paper explores fractal and oscillatory phenomena in the distribution of Goldbach partitions, revealing self-similar patterns and symmetry in prime sums and differences, supported by Hardy-Littlewood estimates and symbolic dynamics analysis.

Contribution

It uncovers fractal structures and periodic oscillations in Goldbach partition statistics, linking them to Hardy-Littlewood estimates and symbolic dynamics.

Findings

Presence of fractal patterns in r(n) series

Oscillations with multiple levels of periodicity

Symmetry between sum and difference distributions

Abstract

Some interesting chaos phenomena have been found in the difference of prime numbers. Here we discuss a theme about the sum of two prime numbers, Goldbach conjecture. This conjecture states that any even number could be expressed as the sum of two prime numbers. Goldbach partition r(n) is the number of representations of an even number n as the sum of two primes. This paper analyzes the statistics of series r(n) (n=4,6,8,...). The familiar 3 period oscillations in histogram of difference of consecutive primes appear in r(n).We also find r(n) series could be divided into different levels period oscillation series. The series in the same or different levels are all very similar, which presents the obvious fractal phenomenon. Moreover, symmetry between the statistics figure of sum and difference of two prime numbers are also described. We find the estimate of Hardy-Littlewood could precisely depict these phenomena. A rough analyzing for periodic behavior of r(n) is given by symbolic dynamics theory at last.