An asymptotic expansion for the generalised quadratic Gauss sum revisited

Abstract

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta )=\sum _{j=1}^N\exp (\pi ix{j}^2+2\pi ij\theta ),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\to 0$ and $N\to \infty$ such that $Nx$ is finite. The form of this expansion holds for all values of $Nx+\theta$ and, in particular, in the neighbourhood of integer values of $Nx+\theta$. 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.

9 pages, no figures

Cited by in corpus (1)

• On Quadratic Gauss Sums and Variations Thereof