Executive Summary : | : In 1918, Ramanujan introduced certain trigonometric sums which are now known as Ramanujan sums. These sums were used to derive pointwise convergent series expansions of various arithmetic functions. This representation is possible due to the remarkable orthogonality property of Ramanujan sums. Several natural questions arise for example which arithmetic functions have Ramanujan representations, if such a representation exists for a function, how do we determine the Ramanujan coefficients, or what can one say about the rate of convergence of such a series. The orthogonality principle allows one to write down possible candidates for the Ramanujan coeffcients of any given arithmetical function. Ramanujan sums and the theory of Ramanujan Fourier Transforms have been used in signal processing, for example in dealing with non-harmonic components in an integer-periodic noisy signal. Objectives : This project proposal is formulated with multiple objectives to develop the theory and study applications of various kinds of the Ramanujan sums, like the Cohen-Ramanujan sums, Ramanujan sums for number fields etc and related Ramanujan expansions. The various objectives of this project are written below including, understanding a possible theory of Ramanujan expansions for the generalized Ramanujan sums, implications of these generalized Ramanujan expansions in the context of some of earlier works of PI on estimating shifted convolution sums of functions with absolutely convergent Ramanujan expansions. Significance : Ramanujan sums have classically been studied by several mathematicians over several years. They have appeared in some of the most fundamental questions related to the understanding of integers. The prominent ones being their usage in the proof of Vinogradov's theorem that says that all sufficiently large odd numbers can be written as a sum of three primes and which itself is a simplified problem originating from the famous Goldbach conjecture. Moreover it showcases itself in Waring type formulas, problems on equipartition modulo odd integers, in the large sieve inequality, and many more. These simple looking functions have found applications in other areas of mathematics as well as outside mathematics, for example in algebra, character theory, combinatorics, graph theory and signal processing. The primary goal of the project is to develop the theory of Ramanujan sums and related Ramanujan expansions in various settings and to see their applications in these various fields. For example, one would like to see the appearance of the generalized Ramanujan sums as weights appearing in the real valued Fourier coefficients of special classes of even and odd-symmetric signals. |