On Ramanujan expansions and primes in arithmetic progressions

TL;DR

This paper explores Ramanujan expansions and their relation to primes in arithmetic progressions, deriving inequalities linked to the Hardy-Littlewood conjecture on twin primes through advanced number theory techniques.

Contribution

It applies Delange's theorem to the von Mangoldt function to establish a new inequality connected to prime distribution and twin primes conjecture.

Findings

Derived an inequality involving prime counting functions in arithmetic progressions

Connected the inequality to the Hardy-Littlewood twin primes conjecture

Extended Ramanujan expansion techniques to prime distribution analysis

Abstract

A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.