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

Wenjia Guo; Yuankui Ma; Tianping Zhang; Wenjia Guo; Yuankui Ma; Tianping Zhang

doi:10.3934/math.2021095

AIMS Mathematics

2021, Volume 6, Issue 2: 1596-1606. doi: 10.3934/math.2021095

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Research article

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

Received: 26 May 2020 Accepted: 23 November 2020 Published: 25 November 2020
MSC : 11F20, 11L05

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 $K(n, p)$.
- 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[J]. AIMS Mathematics, 2021, 6(2): 1596-1606. doi: 10.3934/math.2021095

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Abstract

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 $K(n, p)$.

References

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