Counting fundamental solutions to the Pell equation with prescribed size. (11th November 2018) - Xi, Ping
Abstract : The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^{2}-Du^{2}=1$ is less than $D^{1/2+\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\in [\frac{1}{2}, 1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$ -analogue of van der Corput method to algebraic exponential sums with smooth moduli.