Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-22T00:05:58.807Z Has data issue: false hasContentIssue false
  • English
  • Français

A MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND THE RAMANUJAN'S SUM*

Published online by Cambridge University Press:  09 December 2011

ZHANG WENPENG
ZHANG WENPENG*
Affiliation:
Department of Mathematics, Northwest University, Xi'an, Shaanxi, P.R. China e-mail: wpzhang@nwu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.

Keywords

Primary 11L4011F20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 155 - 162
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Apostol, T. M., Introduction to analytic number theory (Springer-Verlag, New York, 1976).Google Scholar
2.Carlitz, L., The reciprocity theorem of Dedekind sums, Pacific J. Math. 3 (1953), 513–522.CrossRefGoogle Scholar
3.Chaohua, J., On the mean value of Dedekind sums, J. Number Theory 87 (2001), 173–188.Google Scholar
4.Conrey, J. B., Fransen, E., Klein, R. and Scott, C., Mean values of Dedekind sums, J. Number Theory 56 (1996), 214–226.CrossRefGoogle Scholar
5.Funakura, T., On Kronecker's limit formula for Dirichlet series with periodic coefficients, Acta Arith. 55 (1990), 59–73.CrossRefGoogle Scholar
6.Guy, R. K., Unsolved problems in number theory, 2nd edn. (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar
7.Wenpeng, Z., On a problem of D. H. Lehmer and its generalization, Composito Math. 86 (1993), 307–316.Google Scholar
8.Wenpeng, Z., A problem of D. H. Lehmer and its generalization (II), Compositio Math. 91 (1994), 47–56.Google Scholar
9.Wenpeng, Z., A problem of D. H. Lehmer and its mean square value formlua, Jap. J. Math. 29 (2003), 109–116.CrossRefGoogle Scholar
10.Zhang, W., On the mean values of Dedekind sums, J. Theor. Nombres. 8 (1996), 429–442.CrossRefGoogle Scholar
11.Zhang, W., A note on the mean square value of the Dedekind sums, Acta Math. Hung. 86 (2000), 275–289.CrossRefGoogle Scholar