arXiv subjects
Publications and source records attributed to John Washburn.
Using Abel summation the paper proves a weak form of the Wiener-Khinchin formula for arithmetic functions with point-wise convergent Ramanujan-Fourier expansions. The main result is that the convolution of most arithmetic functions possessing an R-F expansion are Abel-summable to a result involving only the Ramanujan-Fourier coefficients of the R-F expansion(s).
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{ϕ(n) Λ\left(n \right)}{n} \frac{ϕ(n+h) Λ\left(n +h\right)}{n+h} = \sum\limits_{q=1}^{\infty} \left\Vert \frac{μ(q)}{ϕ(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.
This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and cross-correlation. The general results for two almost periodic functions are narrowed and applied to Ramanujan sums and finally applied to support the specific relation of the Wiener-Khinchin formula for arithemic functions with a Ramanujan-Fourier Series.