Generalisations of Ramanujan sums for Polynomial rings over finite fields

TL;DR

This paper extends Ramanujan sums to polynomial rings over finite fields, exploring their multiplicative properties, Fourier series, and Dirichlet series, thus broadening the understanding of these sums in algebraic structures.

Contribution

It introduces a generalized form of Ramanujan sums for polynomial rings over finite fields and investigates their key properties and series representations.

Findings

Established multiplicative properties of the generalized sums

Derived finite Fourier series representations

Proved results on the Dirichlet series of these functions

Abstract

In this paper, we consider a general form of the analogue of Ramanujan's sum in the ring of polynomials over a finite field. We first prove some multiplicative properties of such functions before considering their finite Fourier series and some specific examples. In the end, we also prove a result about the Dirichlet series of such functions.