On a Mean Value of Gadiyar and Padma

TL;DR

This paper proves a key limit involving arithmetic functions and Ramanujan-Fourier expansions, linking it to major conjectures like twin primes and Sophie Germain primes, advancing the analytical approach to these longstanding problems.

Contribution

It establishes a new limit theorem connecting Gadiyar and Padma's work with Ramanujan-Fourier series, providing a foundation for addressing prime conjectures.

Findings

Proves a specific limit involving nd unctions.

Links the limit to Ramanujan-Fourier expansions and prime conjectures.

Clarifies convergence properties of the main theorem.

Abstract

Building on the earlier works of Gadiyar and Padma, the main result of this paper is to prove: \begin{equation} \lim_{n \to \infty} \frac{1}{N} \sum_{n=1}^{N} \frac{\phi(n) \Lambda\left(n \right)}{n} \frac{\phi(n+h) \Lambda\left(n +h\right)}{n+h} = \sum\limits_{q=1}^{\infty} \left\Vert \frac{\mu(q)}{\phi(q)} \right\Vert^2 c_q(h) \end{equation} This sieve with Ramanujan-Fourier expansions is the the central relationship to be proven in within the works of H. G. Gadiyar and R. Padma, as related to the following conjectures in number theory: The twinned prime conjecture, The Sophie Germaine Primes conjecture, and Conjectures B and D of Hardy and Littlewood. A reviewer has point out that Theorem 8 from the previous version should be split into two theorems; one for absolute convergence and one for uniform convergence.