Abstract
For 1 ⩾ c ⩾ p − 1, let E 1,E 2, …,E m be fixed numbers of the set {0, 1}, and let a 1, a 2, …, a m (1 ⩽ a i ⩽ p, i = 1, 2, …,m) be of opposite parity with E 1,E 2, …,E m respectively such that a 1 a 2…a m ≡ c (mod p). Let
We are interested in the mean value of the sums
where E(c, m, p) = N(c,m, p)−((p − 1)m−1)/(2m−1) for the odd prime p and any integers m ⩾ 2. When m = 2, c = 1, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.
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This work is supported by Basic Research Fund of Northwestern Polytechnical University of P. R. China (JC201123).
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Ma, R., Zhang, Y. On a kind of generalized Lehmer problem. Czech Math J 62, 1135–1146 (2012). https://doi.org/10.1007/s10587-012-0068-8
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DOI: https://doi.org/10.1007/s10587-012-0068-8