General Kloosterman sums over the ring of Gaussian integers

The general Kloosterman sum K(m, n; k; q) over ℤ was studied by S. Kanemitsu, Y. Tanigawa, Yuan Yi, and Wenpeng Zhang in their research of the problem of D. H. Lehmer. In the present paper, we obtain similar estimates for K(α, β; k; γ) over ℤ[i]. We also consider the sum , which does not have an analog in the ring ℤ but can be used for the investigation of the second moment of the Hecke zeta function of the field ℚ(i). Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1179–1200, September, 2007.