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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)

##### MSC:
 11A07 Congruences; primitive roots; residue systems
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##### References:
 [1] Richard K. Guy , Unsolved Problems in Number Theory , Springer-Verlag, 1981, pp. 139-141. · Zbl 0805.11001 [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 · doi:10.1112/S0025579300002187 [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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