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Analogues of Kloostermans sums. (English. Russian original) Zbl 0897.11028

Izv. Math. 59, No. 5, 971-891 (1995); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 5, 93-102 (1995).
Kloosterman sums are of the type \[ S(a,b;m)=\sum _{1\leq n\leq m, (m,n)=1} e((an^*+bn)/m), \] where \(nn^*\equiv 1\pmod{m}\). The author restricts the sum here to integers \(n=xy\) with \((xy,m)=1\) and \(x\), \(y\) lying in certain intervals. A complicated but perfectly explicit bound is given for such a modified sum. The bilinear shape of the sum is exploited in the proof, as usual in Vinogradov’s method, and the problem is reduced to a pair of linear congruences, one in variables \(y_i\) and the other in the respective inverses \(y_i ^*\).
Reviewer: M.Jutila (Turku)

MSC:

11L05 Gauss and Kloosterman sums; generalizations
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