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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
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