Czechoslovak Mathematical Journal

, Volume 62, Issue 4, pp 1135–1146

# On a kind of generalized Lehmer problem

• Rong Ma
• Yulong Zhang
Article

## Abstract

For 1 ⩾ cp − 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 2a m c (mod p). Let
$$N(c,m,p) = {1 \over {{2^{m - 1}}}}\mathop {\sum\limits_{{a_1} = 1}^{p - 1} {\sum\limits_{{a_2} = 1}^{p - 1} \ldots } }\limits_{{a_1}{a_2} \ldots \equiv c{\rm{ (}}\bmod {\rm{ }}p)} \sum\limits_{{a_m} = 1}^{p - 1} {(1 - {{( - 1)}^{{a_1} + {E_1}}})(1 - {{( - 1)}^{{a_2} + {E_2}}}) \ldots } (1 - {( - 1)^{{a_m} + {E_m}}}).$$
We are interested in the mean value of the sums
$$\sum\limits_{c = 1}^{p - 1} {{E^2}} (c,m,p),$$
where E(c, m, p) = N(c,m, p)−((p − 1) m−1)/(2 m−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.

## Keywords

Lehmer problem character sum Dirichlet L-function asymptotic formula

11N37 11M06

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