Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1135-1146, 2012

On a kind of generalized Lehmer problem

Rong Ma, Yulong Zhang

R. Ma, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, P. R. China, e-mail: marong0109@163.com; Y. Zhang, The School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, P. R. China, e-mail: zzboyzyl@163.com

Abstract: For $1\le c\le p-1$, let $E_1,E_2,\dots,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1, a_2,\dots, a_m$ $(1\le a_i\le p$, $i=1,2,\dots, m)$ be of opposite parity with $E_1,E_2,\dots,E_m$ respectively such that $a_1a_2\dots a_m\equiv c\pmod p$. Let \begin{equation*} N(c,m,p)=\frac1{2^{m-1}}\mathop{\mathop{\sum}_{a_1=1}^{p-1} \mathop{\sum}_{a_2=1}^{p-1}\dots\mathop{\sum}_{a_m=1}^{p-1}} _{a_1a_2\dots a_m\equiv c\pmod p} (1-(-1)^{a_1+E_1})(1-(-1)^{a_2+E_2})\dots(1-(-1)^{a_m+E_m}). \end{equation*} We are interested in the mean value of the sums \begin{equation*} \sum_{c=1}^{p-1}E^2(c,m,p), \end{equation*} where $ E(c,m,p)=N(c,m,p)-({(p-1)^{m-1}})/({2^{m-1}})$ for the odd prime $p$ and any integers $m\ge2$. 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

Classification (MSC 2010): 11N37, 11M06


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