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?



What if the Riemann Hypothesis were false?
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>


