An application of Kloosterman sums. (English) Zbl 0833.11039

Ruzsa, Imre Z.; Schinzel, Andrzej

Compos. Math. 96, No. 3, 323-330 (1995).

For an odd integer \(n\geq 1\) and a positive integer \(a\leq n\) coprime with \(n\) let \(\overline {a}\) denote the unique element of \(A_n:= \{1\leq a\leq n\); \((a, n)= 1\}\) satisfying \(a\overline {a} \equiv 1\bmod n\). The authors prove the following theorem: For every choice of \(\varepsilon= 0,1\) and \(\delta= -1, +1\) we have \[ \bigl|\bigl\{ a\in A_n;\;a-\overline {a} \equiv \varepsilon \bmod 2,\;({\textstyle {a\over n}} )= \delta \bigr\} \bigr|= {\textstyle {{\varphi (n)} \over 4}} c_{n, \delta}+ O \bigl( 2^{v(n)} \sqrt {n} \log^2 n\bigr), \] where \(v(n)\) denotes the number of distinct prime factors of \(n\) and \(c_{n, \delta}= 1+ \delta\) if \(n\) is a perfect square and \(c_{n, \delta} =1\) otherwise. This asymptotic formula improves results of W.-P. Zhang [Compos. Math. 91, 47-56 (1994; Zbl 0798.11001) ] and S. Chaladus [Demonstr. Math. (to appear)].

Reviewer: J.Hinz (Marburg)

MSC:

11L05	Gauss and Kloosterman sums; generalizations
11A07	Congruences; primitive roots; residue systems

Keywords:

parity; asymptotic formula

Citations:

Zbl 0798.11001

References:

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[4]	Terjanian, G. : Letter to A. Schinzel of February 15 , 1993.
[5]	Wenpeng, Zhang : On a problem of D. M. Lehmer and its generalization , Compositio Math. 86 (1993) 307-316. · Zbl 0783.11002