  We establish that three well-known and rather different looking conjectures
about Dirichlet characters and their (weighted) sums, (concerning the
P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible
range in Burgess' estimate for short character sums, and upper bounds for
$L(1,\chi)$ and $L(1+it,\chi)$) are more-or-less "equivalent". We also obtain a
new mean value theorem for logarithmically weighted sums of 1-bounded
multiplicative functions.
