  We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive
Dirichlet character $\chi \pmod q$ with $x\geq q^\epsilon$, which is known to
be true if the Riemann Hypothesis holds for $L(s,\chi)$. We show that it holds
under the weaker assumption that `$100\%$' of the zeros of $L(s,\chi)$ up to
height $\tfrac 14$ lie on the critical line; and establish various other
consequences of having large character sums.
