On exponential sums involving the divisor function over \({\mathbb Z}[i]\)

Abstract. We apply the van der Corput transform to investigate the sums of view \(\sum r(n)g(n)e(f(n)))\), where \(r(n)\) is the number of representations of \(n\) as the sum of two squares of integer numbers. Such sums have been studied by M. Jutila, O. Gunyavy, M. Huxley and etc. Depending on differential properties of the functions \(g(n)\) and \(f(n)\) there have been obtained different kinds of error terms in bounds of the considered sums. In the special case, O. Gunyavy improved the result of M. Jutila in the problem on estimate the exponential sum involving the divisor function \(\tau(n)\). We obtain the asymptotic formula of the sum \(\sum{\tau_k(\alpha)e\left(\frac{a}{q}N(\alpha)\right)}\), \(k=2,3\) over the ring of Gaussian integers which is an analogue of the asymptotic formulas obtained by M. Jutila and O. Gunyavy.