An asymptotic expansion for the generalised quadratic Gauss sum revisited

TL;DR

This paper derives an asymptotic expansion for the generalized quadratic Gauss sum in the limit of small x and large N, with a focus on accuracy and bounds, applicable across all relevant parameter values.

Contribution

It presents a new asymptotic expansion for the generalized quadratic Gauss sum valid as N→∞ and x→0, including a simple remainder bound and numerical validation.

Findings

Expansion valid for all Nx+θ values

Derived a simple bound for the remainder

Numerical results confirm accuracy and sharpness

Abstract

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and $N\rightarrow\infty$ 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.