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arXiv · 1612.08309

On the Polynomial Ramanujan Sums over Finite Fields

Zhiyong Zheng
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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.

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Zhiyong Zheng. 2016-12-26. On the Polynomial Ramanujan Sums over Finite Fields. https://arxiv.org/abs/1612.08309

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