Abstract

The main purpose of this paper is using the properties of Gauss sums and the mean value theorem of the Dirichlet L-functions to study one kind of a hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and to give an exact computational formula for it.MSC:11L05, 11L10.

Highlights

  • Let c be a natural number and d be an integer prime to c

  • The main purpose of this paper is using the properties of Gauss sums and the mean square value theorem of the Dirichlet L-functions to study a hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and to give an exact computational formula for it

  • Authors’ contributions WW carried out the hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and gave an exact computational formula

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Summary

Introduction

Let c be a natural number and d be an integer prime to c. Regarding the properties of S (d, c) and related sums, some authors studied them and obtained many interesting results; see [ – ] and [ ]. The main purpose of this paper is using the properties of Gauss sums and the mean square value theorem of the Dirichlet L-functions to study a hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and to give an exact computational formula for it. Theorem Let q > be an odd square-full number.

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