TITLE: What if the Riemann Hypothesis were false?
QUESTION [66 upvotes]: There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?

REPLY [46 votes]: An explicit zero $\rho$ for $\zeta(s)$, off the critical line, would give an explicit lower bound on the class number $h(-d)$ for $\mathbb Q(\sqrt{-d})$, for a range of $-d$ in terms of $\text{Im}(\rho)$.  This is the 'Deuring-Heilbronn phenomenon,' with results due to these two and others beginning in the 1930's.  For an elementary account, see
http://arxiv.org/abs/1201.0713