In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Polya-Vinogradov inequality. We show how the Dobrowolski-Williams approach is related to the classical proof of the Polya-Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65-71,2001). In passing, we also obtain sharper explicit versions of the the Polya-Vinogradov inequality.
We give an explicit recursive description of the Hilbert series and Gröbner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers–Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.