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Lehmer problem; character sum; Dirichlet $L$-function; asymptotic formula
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)=\frac {1}{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*} \endgraf 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 $m\ge 2$. 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.
[1] Apostol, T. M.: Introduction to Analytic Number Theory. Springer New York (1976). MR 0434929 | Zbl 0335.10001
[2] Guy, R. K.: Unsolved Problems in Number Theory. Springer New York-Heidelberg-Berlin (1981). MR 0656313 | Zbl 0474.10001
[3] Ma, R., Zhang, J., Zhang, Y.: On the $2m$th power mean of Dirichlet $L$-functions with the weight of trigonometric sums. Proc. Indian Acad. Sci., Math. Sci. 119 (2009), 411-421. DOI 10.1007/s12044-009-0046-8 | MR 2647187
[4] Ma, R., Yi, Y., Zhang, Y.: On the mean value of the generalized Dirichlet $L$-functions. Czech. Math. J. 60 (2010), 597-620. DOI 10.1007/s10587-010-0056-9 | MR 2672404 | Zbl 1224.11077
[5] Xu, Z., Zhang, W.: On the $2k$th power mean of the character sums over short intervals. Acta Arith. 121 (2006), 149-160. DOI 10.4064/aa121-2-4 | MR 2216139 | Zbl 1153.11046
[6] Xu, Z., Zhang, W.: On a problem of D. H. Lehmer over short intervals. J. Math. Anal. Appl 320 (2006), 756-770. DOI 10.1016/j.jmaa.2005.07.054 | MR 2225991 | Zbl 1098.11050
[7] Zhang, W.: On a problem of D. H. Lehmer and its generalization. Compos. Math. 86 (1993), 307-316. MR 1219630 | Zbl 0783.11003
[8] Zhang, W.: A problem of D. H. Lehmer and its generalization (II). Compos. Math. 91 (1994), 47-56. MR 1273925 | Zbl 0798.11001
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