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Some new sums related to D. H. Lehmer problem

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Abstract

About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N(k; p) denote the number of all 1 ⩽ a i ⩽ p − 1 such that a 1 a 2 … a k ≡ 1 mod p and 2 | a i + ā i + 1, i = 1, 2, …, k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N(k; p), and give an interesting asymptotic formula for it.

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Authors and Affiliations

  1. School of Mathematics, Northwest University, Xuefu Avenue No. 1, Chang’an, Xi’an, Shaanxi, 710127, P. R. China

    Han Zhang & Wenpeng Zhang

Authors
  1. Han Zhang
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  2. Wenpeng Zhang
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Correspondence to Han Zhang.

Additional information

This work was supported by the P. S. F. (2013JZ001) and N. S. F. (11371291) of P. R. China.

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Zhang, H., Zhang, W. Some new sums related to D. H. Lehmer problem. Czech Math J 65, 915–922 (2015). https://doi.org/10.1007/s10587-015-0217-y

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  • DOI: https://doi.org/10.1007/s10587-015-0217-y

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