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:; Y. Zhang, The School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, P. R. China, e-mail:

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

Full text available as PDF.

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]