  The behavior of harmonic functions on Riemannian manifolds under lower bounds
of the Ricci curvature has been studied from both analytic and geometric
viewpoints. For example, some Liouville type theorems are obtained under lower
bounds of the Ricci curvature. Recently, those results are generalized under
lower bounds of the $N$-weighted Ricci curvature $\mbox{Ric}_\psi^N$ with $N
\in [n,\infty]$. In this paper, we present a Liouville type theorem for
harmonic functions of sublinear growth and a gradient estimate of harmonic
functions on weighted Riemannian manifolds under weaker lower bounds of
$\mbox{Ric}_\psi^N$ with $N < 0$. We also prove an $L^p$-Liouville theorem
under lower bounds of $\mbox{Ric}_\psi^N$ with $N \in [n,\infty]$ in a way
different from that of [Wu, 2014]. Our results are obtained as a consequence of
an argument under lower bounds of $\mbox{Ric}_\psi^N$ with $\varepsilon$-range,
which is a unification of constant and variable curvature bounds. Among various
methods considered for the analysis of harmonic functions, this paper focuses
on methods using the Moser's iteration procedure.
