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.
Categories: math.GM