AIMS Mathematics, 2021, 6(2): 1596-1606. doi: 10.3934/math.2021095.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

New identities involving Hardy sums $S_3(h,k)$ and general Kloosterman sums

1 School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, P. R. China
2 School of Science, Xi’an Technological University, Xi’an, Shaanxi, P. R. China

The main purpose of this paper is to obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums $S_{3}(h,p)$ and general Kloosterman sums $K(r,l,\lambda;p)$. By applying the properties of Gauss sums and the mean value theorems of Dirichlet $L$-function, we derive some new identities. As the special cases, we also deduce some exact computational formulas for hybrid mean value involving $S_{3}(h,p)$ and classical Kloosterman sums $K(n,p)$.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Hardy sums; general Kloosterman sums; hybrid mean value

Citation: Wenjia Guo, Yuankui Ma, Tianping Zhang. New identities involving Hardy sums $S_3(h,k)$ and general Kloosterman sums. AIMS Mathematics, 2021, 6(2): 1596-1606. doi: 10.3934/math.2021095

References

  • 1. T. M. Apostol, Modular function and Dirichlet series in number theory, New York: Springer-Verlag, 1976.
  • 2. L. Carlitz, The reciprocity theorem of Dedekind sums, Pacific J. Math., 3 (1953), 513-522.
  • 3. J. B. Conrey, E. Fransen, R. Klein, C. Scott, Mean values of Dedekind sums, J. Number Theory, 56 (1996), 214-226.
  • 4. X. L. He, W. P. Zhang, On the mean value of the Dedekind sum with the weight of Hurwitz zeta-function, J. Math. Anal. Appl., 240 (1999), 505-517.
  • 5. B. C. Berndt, L. A. Goldberg, Analytic properties of arithmetic sums arising in the theory of the classical theta-function, SIAM J. Math. Anal., 15 (1984), 143-150.
  • 6. H. Zhang, W. P. Zhang, On the identity involving certain Hardy sums and Kloosterman sums, Inequal. Appl., 52 (2014), 1-9.
  • 7. H. F. Zhang, T. P. Zhang, Some identities involving certain Hardy sums and general Kloosterman sums, Mathematics, 8 (2020), 95.
  • 8. B. C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, Reine Angew. Math., 303-304 (1978), 332-365.
  • 9. L. A. Goldberg, Transformations of theta-functions and analogues of Dedekind sums, Ph.D. thesis, University of Illinois, Urbana, 1981.
  • 10. R. Sitaramachandrarao, Dedekind and Hardy sums, Acta Arith., 48 (1978), 325-340.
  • 11. W. P. Zhang, On the mean values of Dedekind sums, J. Theor. Nombr. Bordx., 8 (1996), 429-442.

 

Reader Comments

your name: *   your email: *  

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved