×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: Numdam EuDML
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.