On the distribution of values of Hardy's $Z$-functions in short intervals, II: The $q$-aspect

Abstract

We continue our investigations regarding the distribution of positive and negative values of Hardy’s $Z$-functions $Z\left(t,\chi \right)$ in the interval $\left[T,T+H\right]$ when the conductor $q$ and $T$ both tend to infinity. We show that for $q\le T^{\eta }$, $H=T^{\vartheta }$, with $\vartheta >0$, $\eta >0$ satisfying $\frac{1}{2}+\frac{1}{2}\eta <\vartheta \le 1$, the Lebesgue measure of the set of values of $t\in \left[T,T+H\right]$ for which $Z\left(t,\chi \right)>0$ is $\gg \left(\phi {\left(q\right)}^2∕4^{\omega \left(q\right)}{q}^2\right)H$ as $T\to \infty$, where $\omega \left(q\right)$ denotes the number of distinct prime factors of the conductor $q$ of the character $\chi$, and $\phi$ is the usual Euler totient. This improves upon our earlier result. We also include a corrigendum for the first part of this article.