On the Polynomial Ramanujan Sums over Finite Fields

TL;DR

This paper explores the arithmetic and analytic properties of polynomial Ramanujan sums over finite fields, establishing fundamental identities and analyzing associated Dirichlet series, revealing their entire function nature and mean value estimates.

Contribution

It introduces new identities and properties of polynomial Ramanujan sums, including their entire Dirichlet series and mean value bounds, expanding understanding of these sums over finite fields.

Findings

Polynomial Ramanujan sums satisfy Holder, reciprocity, and orthogonality formulas.

Dirichlet series involving these sums are entire functions on the complex plane.

Square mean value estimates for the sums are established.

Abstract

The polynomial Ramanujan sum was first introduced by Carlitz [7], and a generalized version by Cohen [10]. In this paper, we study the arithmetical and analytic properties of these sums, derive various fundamental identities, such as H older formula, reciprocity formula, orthogonality relation and Davenport{Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, we also give a square mean values estimation. The main results of this paper are new appearance to us, which indicate the particularity of the polynomial Ramanujan sums.