On the distribution and applications of Ramanujan sums

Abstract

While studying the trigonometric series expansion of certain arithmetic functions, Ramanujan, in 1918, defined a sum of the nth power of the primitive qth roots of unity and denoted it as cq(n). These sums are now known as Ramanujan sums. Since then, Ramanujan sums have been widely used and studied in mathematics and other areas. Most importantly, it is used in the proof of Vinogradov’s theorem that every sufficiently large odd number is the sum of three primes. It is also used to simplify the computations of Arithmetic Fourier Transform (AFT), Discrete Fourier Transform (DFT), and Discrete Cosine Transform (DCT) coefficients for a special type of signal. We study Ramanujan sums in the context of the k-tuple prime conjecture. A twin prime is a prime number that is either two less or two more than another prime number. It is conjectured that there are infinitely many twin primes. Hardy and Littlewood generalized the twin prime conjecture and gave the k-tuple conjecture. Let d1, · · · , dk be distinct integers, and b(p) is the number of distinct residue classes (mod p) represented by di . If b(p) < p for every prime p, the k-tuple conjecture gives an asymptotic formula for the number of n ≤ x such that all the k numbers n + di are primes. We study a heuristic proof of the k-tuple conjecture using the convolution of Ramanujan sums. Additionally, we study questions on the distribution of Ramanujan sums. One way to study distribution is via moments of averages. Chan and Kumchev studied the first and second moments of Ramanujan sums. In this thesis, we estimate the higher moment of their averages using the theory of functions of several variables initiated by Vaidyanathaswamy. Ramanujan sums can also be generalized over number fields. A number field is an extension field K of the field of rational numbers Q such that the field extension K/Q has a finite degree. Nowak first studied the first moment for Ramanujan sums over quadratic number fields, and later, it was estimated for the higher degree number fields as well. For a general number field, assuming generalized Lindelöf Hypothesis, we improve the first moment result and also study the second moment. Furthermore, unconditionally, we estimate asymptotic formulas for the second moment for quadratic, cubic, and cyclotomic number fields. Our primary tool for these results is a Perron-type formula. Finally, we obtain the second moment result for certain integral domains called Prüfer domains.