Kloosterman Sums and a Problem of D. H. Lehmer

Citation:

Ping XI,Yuan YI.Kloosterman Sums and a Problem of D. H. Lehmer[J].Chinese Annals of Mathematics B,2020,41(3):361~370
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Authors:

Ping XI; Yuan YI

Foundation:

This work was supported by the National Natural Science Foundation of China (No.11601413), the Fundamental Research Funds for the Central Universities (No.201806078) and the Natural Science Basic Research Plan in Shaanxi Province of China (No.2017JQ1016).
Abstract: A classical problem of D. H. Lehmer suggests the study of distributions of elements of $\bZ/p\bZ$ of opposite parity to the multiplicative inverse mod $p$. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved, from which one may see that Zhang's error term is optimal up to the logarithm factor. The method also applies to the case of arbitrary positive integral moments.

Keywords:

D. H. Lehmer problem, Kloosterman sum, Moment

Classification:

11N69, 11L05
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