## A mean value related to the D. H. Lehmer problem and Kloosterman sums.(English)Zbl 1220.11123

For $$q>2$$, $$(c,q)=1$$, $$(a,q)=1$$, $$1\leq a\leq q-1$$, there exists $$b$$, $$1\leq b\leq q-1$$ such that $$ab\equiv c\bmod q$$.
Let $$N(c,q)$$ denote the number of cases in which $$a,b$$ are of opposite parity, $$K(m,n;q)$$ be the Kloosterman sums, and $$E(c,q)=N(c,q)-\varphi(q)/2$$. The authors obtain: $\sum_{c=1}^{q} K(c,1;q)E(4c,q)=\frac{4}{\pi^{2}}q\varphi(q)\prod\limits_{p\| q}\left(1-\frac{1}{p(p-1)}\right)+O\left(q\,\exp\left(\frac{13\ln q}{\ln\ln q}\right)\right).$
The proof uses the asymptotic formula for $$L^{2}(1,\chi)$$ and the Gauss sums.

### MSC:

 11N37 Asymptotic results on arithmetic functions 11L05 Gauss and Kloosterman sums; generalizations

### Keywords:

Kloosterman sums; hybrid mean value; asymptotic formula
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