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?
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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 'DeuringHeilbronn phenomenon,' with results due to these two and others beginning in the 1930's. For an elementary account, see 

