Convolution and Cross-correlation of Ramanujan-fourier Series

where x̄ is the complex conjugate of x 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 (1). The Wiener-Khinchin fomula, (1), connecting the auto-correlation of an arithmetic function and the coefficients of its Ramanujan Fourier series is a powerful link between the circle method and the sieve methods found in number theory. The application of this Weiner-Khinchin formula to number theory is described in the works of H. G. Gadiyar and R. Padma [4], [5], [6], [7]. The Wiener-Khinchin fomula is used in [4] to prove the Hardy-Littlewood conjecture and is used in [6] to prove the density of Sophie primes.