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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 $x\rightarrow0$ and $N\rightarrow\infty$ such that $Nx$ is finite. The form of thisexpansion holds for all values of $Nx+\theta$ and, in particular, in theneighbourhood of integer values of $Nx+\theta$. A simple bound for theremainder in the expansion is derived. Numerical results are presented todemonstrate the accuracy of the expansion and the sharpness of the bound.

An asymptotic expansion for the generalised quadratic Gauss sum revisited