Zhang, Wenpeng A problem of D. H. Lehmer and its generalization. II. (English) Zbl 0798.11001 Compos. Math. 91, No. 1, 47-56 (1994). For an odd integer \(q>2\) let \(L(q)\) denote the set of integers \(a\) in the interval \([1,q-1]\) that are coprime with \(q\) and for which \(a + \overline a \equiv 1 \text{mod} 2\) holds. Here, \(\overline a\) stands for the solution of the congruence \(ax \equiv 1 \text{mod} q\) with \(1 \leq \overline a \leq q-1\). In [Chin. Sci. Bull. 38, No. 16, 1340-1345 (1993; Zbl 0797.11004)] Z. Zheng derived for \(1 \leq N \leq q-1\) the asymptotic formula \[ \sum^ N_{ {a = 1 \atop a \in L(q)}}1 = {N \over 2} \cdot {\varphi (q) \over q} + O \bigl( q^{1/2} d(q) \log^ 2q \bigr), \] where \(d(q)\) is the divisor function. In the present paper the author also proves this result in the case \(N=q-1\). Reviewer: J.Hinz (Marburg) Cited in 7 ReviewsCited in 33 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11N69 Distribution of integers in special residue classes 11A15 Power residues, reciprocity Keywords:parity; Lehmer’s problem Citations:Zbl 0797.11004 PDFBibTeX XMLCite \textit{W. Zhang}, Compos. Math. 91, No. 1, 47--56 (1994; Zbl 0798.11001) Full Text: Numdam EuDML References: [1] Richard K. Guy, Unsolved Problems in Number Theory , Springer-Verlag, 1981, pp. 139-140. · Zbl 0805.11001 [2] Zhang Wenpeng , On a problem of D. H. Lehmer and its generalization , Compositio Mathematica. 86 (1993) 307-316. · Zbl 0783.11002 [3] Funakura, Takeo , On Kronecker’s limit formula for Dirichlet series with periodic coefficients , Acta Arith., 55 (1990), No. 1, pp. 59-73. · Zbl 0654.10039 [4] T. Estermann , On Kloostermann’s sum , Mathematica, 8 (1961), pp. 83-86. · Zbl 0114.26302 · doi:10.1112/S0025579300002187 [5] Apostol, Tom M , Introduction to Analytic Number Theory , Springer-Verlag, New York, 1976. · Zbl 0335.10001 [6] J.-M. Deshouillers and H. Iwaniec , Kloosterman Sums and Fourier coefficients of Cusp Forms , Inventiones Mathematicae, 70 (1982) pp. 219-288. · Zbl 0502.10021 · doi:10.1007/BF01390728 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.