On the Polynomial Ramanujan Sums over Finite Fields

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.

arXiv comment29 pages

References in corpus (3)

• Ramanujan sums for signal processing of low frequency noise
• Ramanujan sums analysis of long-period sequences and 1/f noise
• Some remarks on Ramanujan sums and cyclotomic polynomials