Advertisement

Czechoslovak Mathematical Journal

, Volume 62, Issue 4, pp 1135–1146 | Cite as

On a kind of generalized Lehmer problem

  • Rong Ma
  • Yulong Zhang
Article
  • 75 Downloads

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 

MSC 2010

11N37 11M06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. M. Apostol: Introduction to Analytic Number Theory. Springer, New York, 1976.MATHGoogle Scholar
  2. [2]
    R. K. Guy: Unsolved Problems in Number Theory. Springer, New York-Heidelberg-Berlin, 1981.MATHGoogle Scholar
  3. [3]
    R. Ma, J. Zhang, Y. Zhang: On the 2mth power mean of Dirichlet L-functions with the weight of trigonometric sums. Proc. Indian Acad. Sci., Math. Sci. 119 (2009), 411–421.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    R. Ma, Y. Yi, Y. Zhang: On the mean value of the generalized Dirichlet L-functions. Czech. Math. J. 60 (2010), 597–620.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Z. Xu, W. Zhang: On the 2kth power mean of the character sums over short intervals. Acta Arith. 121 (2006), 149–160.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Z. Xu, W. Zhang: On a problem of D.H. Lehmer over short intervals. J. Math. Anal. Appl 320 (2006), 756–770.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    W. Zhang: On a problem of D.H. Lehmer and its generalization. Compos. Math. 86 (1993), 307–316.MATHGoogle Scholar
  8. [8]
    W. Zhang: A problem of D.H. Lehmer and its generalization (II). Compos. Math. 91 (1994), 47–56.MATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.School of ScienceNorthwestern Polytechnical UniversityXi’an, ShaanxiP.R.China
  2. 2.The School of Electronic and Information EngineeringXi’an Jiaotong UniversityXi’an, ShaanxiP.R.China

Personalised recommendations