An asymptotic expansion for the generalised quadratic Gauss sum revisited

الأصل · EN

An asymptotic expansion for the generalised quadratic Gauss sum Sₙ(x,θ)=∑ⱼ₌₁ⁿ (πixj²+2πijθ), where x, θ are real and N is a positive integer, is obtained as x→ 0 and N→∞ such that Nx is finite. The form of this expansion holds for all values of Nx+θ and, in particular, in the neighbourhood of integer values of Nx+θ. A simple bound for the remainder in the expansion is derived. Numerical results are presented to demonstrate the accuracy of the expansion and the sharpness of the bound.