On a problem of D. H. Lehmer and its generalization. (English) Zbl 0783.11002

Let \(q>1\) be an odd integer. Let \(r(q)\) denote the number of integers in the interval \([1,q-1]\) that are coprime with \(q\) and for which \(x\) and \(\overline{x}\) are of opposite parity, i.e. \(x+\overline{x}\equiv 1\bmod 2\). The element \(\overline{x}\) is given by \(\overline{x} x\equiv 1\bmod q\) with \(0<\overline{x}<q\). In this paper the author derives asymptotic formulae for \(r(q)\) with \(q=p^ \alpha\) or \(q=pp'\), \(p,p'\in\mathbb{P}\), \(p\neq p'\), \(\alpha\in\mathbb{N}\) (see also the following reviews). The proof makes use of estimates for character sums and Kloosterman sums.
Reviewer: J.Hinz (Marburg)


11A07 Congruences; primitive roots; residue systems


Zbl 0783.11004
Full Text: Numdam EuDML


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[2] G. Pólya , Über die Verteilung der quadratischen Reste und Nichtreste , Göttinger Nachrichten, 1918, pp. 21-29. · JFM 46.0265.02
[3] T. Estermann , On Kloosterman’s sum , Mathematika, 8 (1961), pp. 83-86. · Zbl 0114.26302
[4] Apostol, Tom M. , Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics , Springer-Verlag, New York, 1976. · Zbl 0335.10001
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