On a kind of generalized Lehmer problem

Rong Ma; Yulong Zhang

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 4, page 1135-1146
  • ISSN: 0011-4642

Abstract

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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 N ( c , m , p ) = 1 2 m - 1 ∑ a 1 = 1 p - 1 ∑ a 2 = 1 p - 1 ⋯ ∑ a m = 1 p - 1 a 1 a 2 ⋯ a m ≡ c ( mod p ) ( 1 - ( - 1 ) a 1 + E 1 ) ( 1 - ( - 1 ) a 2 + E 2 ) ⋯ ( 1 - ( - 1 ) a m + E m ) . We are interested in the mean value of the sums ∑ 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.

How to cite

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Ma, Rong, and Zhang, Yulong. "On a kind of generalized Lehmer problem." Czechoslovak Mathematical Journal 62.4 (2012): 1135-1146. <http://eudml.org/doc/246636>.

@article{Ma2012,
abstract = {For $1\le c\le p-1$, let $E_1,E_2,\dots ,E_m$ be fixed numbers of the set $\lbrace 0,1\rbrace $, 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 $a_1a_2\dots a_m\equiv c\hspace\{4.44443pt\}(\@mod \; p)$. 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\hspace\{10.0pt\}(\@mod \; 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 $m\ge 2$. 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.},
author = {Ma, Rong, Zhang, Yulong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lehmer problem; character sum; Dirichlet $L$-function; asymptotic formula; Lehmer problem; character sum; Dirichlet -function; asymptotic formula},
language = {eng},
number = {4},
pages = {1135-1146},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a kind of generalized Lehmer problem},
url = {http://eudml.org/doc/246636},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Ma, Rong
AU - Zhang, Yulong
TI - On a kind of generalized Lehmer problem
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 1135
EP - 1146
AB - For $1\le c\le p-1$, let $E_1,E_2,\dots ,E_m$ be fixed numbers of the set $\lbrace 0,1\rbrace $, 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 $a_1a_2\dots a_m\equiv c\hspace{4.44443pt}(\@mod \; p)$. 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\hspace{10.0pt}(\@mod \; 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 $m\ge 2$. 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.
LA - eng
KW - Lehmer problem; character sum; Dirichlet $L$-function; asymptotic formula; Lehmer problem; character sum; Dirichlet -function; asymptotic formula
UR - http://eudml.org/doc/246636
ER -

References

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  8. Zhang, W., A problem of D. H. Lehmer and its generalization (II), Compos. Math. 91 (1994), 47-56. (1994) Zbl0798.11001MR1273925

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