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The Ramanujan Journal

, Volume 46, Issue 3, pp 863–898 | Cite as

On the polynomial Ramanujan sums over finite fields

  • Zhiyong Zheng
Article

Abstract

The polynomial Ramanujan sum was first introduced by Carlitz (Duke Math J 14:1105–1120, 1947), and a generalized version by Cohen (Duke Math J 16:85–90, 1949). In this paper, we study the arithmetical and analytic properties of these sums, deriving various fundamental identities, such as Hölder 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, and 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.

Keywords

Polynomial Ramanujan sums Finite fields Reciprocity formula Orthogonality relation Davenport–Hasse’s type formula 

Mathematics Subject Classification

Primary 11T55 11T24 Secondary 11L05 

Notes

Acknowledgements

The author would like to thank the referees for their very careful reading on this paper and pointing out a mistake in the main theorems.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang UniversityBeijingPeople’s Republic of China

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