On a kind of Dirichlet character sums. (English) Zbl 1470.11246

Summary: Let \(p \geq 3\) be a prime and let \(\chi\) denote the Dirichlet character modulo \(p\). For any prime \(q\) with \(q < p\), define the set \(E \left(q, p\right) = \left\{a \mid 1 \leq a, \overline{a} \leq p, a \overline{a} \equiv 1 \pmod{p} \text{ and } a \equiv \overline{a} \pmod{q}\right\}\). In this paper, we study a kind of mean value of Dirichlet character sums \(\sum a \leq p \, a \in E \left(q, p\right) \chi(a)\), and use the properties of the Dirichlet \(L\)-functions and generalized Kloosterman sums to obtain an interesting estimate.


11N37 Asymptotic results on arithmetic functions
11A07 Congruences; primitive roots; residue systems
11L05 Gauss and Kloosterman sums; generalizations
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