A NEW MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND KLOOSTERMAN SUMS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 52, Issue 1, 2015, pp.35-43
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2015.52.1.035

Title & Authors

A NEW MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND KLOOSTERMAN SUMS

Han, Di; Zhang, Wenpeng;

Han, Di; Zhang, Wenpeng;

Abstract

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with , it is clear that the exists one and only one b with such that (mod q). Let N(c, q) denote the number of all solutions of the congruence equation (mod q) for , in which a and are of opposite parity, where is defined by the congruence equation (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 and Kloosterman sums, and give a sharper asymptotic formula for it.

Keywords

D. H. Lehmer's problem;error term;Kloosterman sums;hybrid mean value;asymptotic formula;

Language

English

References

1.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.

2.

D. A. Burgess, On Dirichlet characters of polynomials, Proc. London Math. Soc. 13 (1963), 537-548.

3.

T. Funakura, On Kronecker's limit formula for Dirichlet series with periodic coefficients, Acta Arith. 55 (1990), no. 1, 59-73.

4.

R. K. Guy, Unsolved Problems in Number Theory, Second Edition, Springer-Verlag, New York, 1994.

5.

H. Walum, An exact formula for an average of L-series, Illinois J. Math. 26 (1982), no. 1, 1-3.

6.

W. Zhang, On a problem of D. H. Lehmer and its generalization, Compositio Math. 86 (1993), no. 3, 307-316.

7.

W. Zhang, A note on the mean square value of the Dedekind sums, Acta Math. Hungar. 86 (2000), no. 4, 275-289.