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

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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

Received: , Accepted: , Published:

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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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