Sums of products of Ramanujan sums

TL;DR

This paper explores complex arithmetic functions involving products of Ramanujan sums, deriving new orthogonality relations and expanding understanding of their properties in number theory.

Contribution

It introduces new sums of products of Ramanujan sums with polynomial arguments and establishes a modified orthogonality relation for these sums.

Findings

Derived a modified orthogonality relation for Ramanujan sums

Analyzed arithmetic functions involving products of Ramanujan sums with polynomial arguments

Expanded the theoretical framework of Ramanujan sums in number theory

Abstract

The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$, where $g_1,..., g_r$ are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.