The Kloosterman sums on the ellipse

Abstract

The main point of our research is to obtain the estimates for Kloosterman sums \(\widetilde{K}(\alpha,\beta;h,q;k)\) considered on the ellipse bound for the case of the integer rational module \(q\) and for some natural number \(k\) with conditions \((\alpha,q)=(\beta,q)=1\) on the integer numbers of imaginary quadratic field. These estimates can be used to construct the asymptotic formulas for the sum of divisors function \(\tau_\ell(\alpha)\) for \(\ell=2,3,\ldots\) over the ring of integer elements of imaginary quadratic field in arithmetic progression.