An application of Kloosterman sums. (English) Zbl 0833.11039

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)


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


Zbl 0798.11001
Full Text: Numdam EuDML


[1] Cha\?Adus, S. : An application of Nagell’s estimate for the least quadratic non-residue , Demonstratio Math., to appear. · Zbl 0843.11003
[2] Hasse, H. : Vorlesungen über Zahlentheorie , Berlin, 1950. · Zbl 0038.17703
[3] Malyshev, A.V. : A generalization of Kloosterman sums and their estimates (in Russian) , Vestnik Leningrad. Univ. 15 (1960) No. 13, vypusk 3, 59-75. · Zbl 0102.03801
[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
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.