Distribution of Primes and of Interval Prime Pairs Based on $\Theta$ Function

TL;DR

This paper introduces a new $ heta$ function based on the Kronecker symbol to analyze prime and composite distributions, explores the Goldbach Conjecture's structure, and discusses the lower bounds of prime pair intervals using Abel's Theorem.

Contribution

It proposes a novel $ heta$ function linked to prime distribution and provides an analytical framework for Goldbach Conjecture interval distribution and prime pair lower bounds.

Findings

$ heta$ function effectively models prime/composite distribution

Interval distribution of prime pairs related to Goldbach Conjecture analyzed

Lower bounds of prime pair intervals discussed using Abel's Theorem

Abstract

$\Theta$ function is defined based upon Kronecher symbol. In light of the principle of inclusion-exclusion, $\Theta$ function of sine function is used to denote the distribution of composites and primes. The structure of Goldbach Conjecture has been analyzed, and $\Xi$ function is brought forward by the linear diophantine equation; by relating to $\Theta$ function, the interval distribution of composite pairs and prime pairs (i.e. the Goldbach Conjecture) is thus obtained. In the end, Abel's Theorem (Multiplication of Series) is used to discuss the lower limit of the distribution of the interval prime pairs.