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.

29 pages

References in corpus (2)

• Ramanujan sums analysis of long-period sequences and 1/f noise
• Some remarks on Ramanujan sums and cyclotomic polynomials