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
$\gg \left(\phi {\left(q\right)}^{2}\u2215{4}^{\omega \left(q\right)}{q}^{2}\right)H$
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.
