On sums of generalized Ramanujan sums

TL;DR

This paper explores properties of generalized Ramanujan sums across various algebraic structures, establishing relationships with Dedekind zeta function residues and providing examples in number fields.

Contribution

It introduces new properties of generalized Ramanujan sums and links them to Dedekind zeta function residues, extending previous work to broader algebraic contexts.

Findings

Derived relational expressions between Ramanujan sums and Dedekind zeta residues

Provided examples of generalized sums in number fields

Extended properties of Ramanujan sums to semigroups and quadratic fields

Abstract

Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.