## 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)

### MSC:

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

### Keywords:

parity; asymptotic formula

Zbl 0798.11001
Full Text:

### References:

  Cha\?Adus, S. : An application of Nagell’s estimate for the least quadratic non-residue , Demonstratio Math., to appear. · Zbl 0843.11003  Hasse, H. : Vorlesungen über Zahlentheorie , Berlin, 1950. · Zbl 0038.17703  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  Terjanian, G. : Letter to A. Schinzel of February 15 , 1993.  Wenpeng, Zhang : On a problem of D. M. Lehmer and its generalization , Compositio Math. 86 (1993) 307-316. · Zbl 0783.11002
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