Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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

  • Han, Di (School of Mathematics Northwest University) ;
  • Zhang, Wenpeng (School of Mathematics Northwest University)
  • Received : 2012.06.27
  • Published : 2015.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with $1{\leq}a{\leq}q-1$, it is clear that the exists one and only one b with $0{\leq}b{\leq}q-1$ such that $ab{\equiv}c$ (mod q). Let N(c, q) denote the number of all solutions of the congruence equation $ab{\equiv}c$ (mod q) for $1{\leq}a$, $b{\leq}q-1$ in which a and $\bar{b}$ are of opposite parity, where $\bar{b}$ is defined by the congruence equation $b\bar{b}{\equiv}1$ (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 $(N(c,q)-\frac{1}{2}{\phi}(q))$ and Kloosterman sums, and give a sharper asymptotic formula for it.

Keywords

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. https://doi.org/10.4064/aa-55-1-59-73
  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. https://doi.org/10.1023/A:1006724724840
  8. W. Zhang, Moments of generalized quadratic Gauss sums weighted by L-functions, J. Number Theory 92 (2002), no. 2, 304-314. https://doi.org/10.1006/jnth.2001.2715
  9. W. Zhang, A mean value related to D. H. Lehmer's problem and the Ramanujan's sum, Glasg. Math. J. 54 (2012), no. 1, 155-162. https://doi.org/10.1017/S0017089511000498