Fourth power mean of the general 4-dimensional Kloosterman sum mod p

Abstract

In this article, we prove an asymptotic formula for the fourth power mean of a general 4-dimensional Kloosterman sum. We use a result of P. Deligne, which counts the number of \(\mathbb {F}_p\)-points on the surface

$$\begin{aligned} (x-1)(y-1)(z-1)(1-xyz)-uxyz=0, ~ u\ne 0, \end{aligned}$$

and then take an average of the error terms over u to prove the asymptotic formula. We also find the number of solutions of certain congruence equations mod p which are used to prove our main result.

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Acknowledgements

We are grateful to Professor P. Deligne for his proof of Lemma 4 and his valuable comments on the article. We thank the anonymous referee for his/her thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the paper. We also thank Professor Todd Cochrane for his valuable comments which helped us to improve the main result of this article. The second author is partially supported by a research grant under the MATRICS scheme of SERB, Department of Science and Technology, Government of India.

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Correspondence to Nilanjan Bag.

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Bag, N., Barman, R. Fourth power mean of the general 4-dimensional Kloosterman sum mod p. Res. number theory 6, 31 (2020). https://doi.org/10.1007/s40993-020-00206-0

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Keywords

  • The general s-dimensional Kloosterman sums
  • Dirichlet character
  • Asymptotic formula

Mathematics Subject Classification

  • 11L05