The asymptotic distribution of Kloosterman sums

Abstract

In this paper we shall discuss the asymptotic distribution of a wide class of generalized Kloosterman sums.To define these we let k be a global field and S a finite set of places of k containing the infinite ones, if there are any.Let k S = v∈S k v where k v denotes, as usual, the completion of k at v. Let R be the ring of S-integers of k; it is a discrete, cocompact subring of k S .Let n ∈ N and µ n (k) = {ζ ∈ k : ζ n = 1}.We shall assume that µ n (k) has n elements and that all the divisors of n lie in S.This means that n is invertible in R. Let e : k S → C × be a non-trivial additive character, trivial on R. We denote the fractional ideal {x ∈ k : e|xR = 1} by d(e) -1 ; then d(e) is an ideal of R.Letn denote the nth order Legendre symbol in R. For a,

Locations

• Acta Arithmetica - View - PDF