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

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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

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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 $K(n,p)$.
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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.