On a Mean Value of Gadiyar and Padma

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.