The Kloosterman sums on the ellipse

Abstract

The main point of our research is to obtain the estimates for Kloosterman sums K(α, β; h, q; k) considered on the ellipse bound for the case of the integer rational moduleq and forsome natural number k with conditions (α, q)=(β, q)=1 on the integer numbers of imaginary quadratic field. These estimates can be used to construct the asymptotic formulas for the sum of divisors function τℓ(α)forℓ= 2,3, . . . over the ring of integer elements of imaginary quadratic field in arithmetic progression.