Spectral Methods And Prime Numbers Counting Problems

TL;DR

This paper develops a rigorous spectral method to prove the dePolignac conjecture, establishing the existence of infinitely many prime pairs separated by 2k, advancing the understanding of prime distribution.

Contribution

It introduces a rigorous spectral approach to prove the dePolignac conjecture, extending heuristic spectral analysis to a formal proof.

Findings

Proof of the dePolignac conjecture for all k ≥ 1

Validation of spectral methods in prime number theory

Advancement towards proving the twin prime conjecture

Abstract

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof of the more general dePolignac conjecture on the existence of infinitely many primes pairs p and p + 2k, k => 1, is proposed in this note.