An asymptotic expansion for the generalised quadratic Gauss sum revisited

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.