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A problem of D. H. Lehmer and its generalization. II. (English) Zbl 0798.11001
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)

##### MSC:
 11A07 Congruences; primitive roots; residue systems 11N69 Distribution of integers in special residue classes 11A15 Power residues, reciprocity
##### Keywords:
parity; Lehmer’s problem
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##### 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 · numdam:CM_1993__86_3_307_0 · eudml:90223 [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 · eudml:206269 [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 · eudml:142975
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