# What if the Riemann Hypothesis were false?

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?

• If it were false, a consequence would be that the distribution of the primes would have be to be more interesting than currently (generally) believed. This is a bit of a meta answer. But it would be highly interesting if it were false. In that sense RH true is the more "boring" case.
– user9072
Jul 11, 2013 at 15:11
• In the early 20th century, the proof that the class number of imaginary quadratic fields ${\mathbf Q}(\sqrt{-d})$ for squarefree $d > 0$ tends to $\infty$ as $d \rightarrow \infty$ was based on a two-part argument: Landau showed that it follows from the assumption that GRH is true for the $L$-functions of all imaginary quadratic Dirichlet characters, and then Heilbronn showed that it follows from the assumption that GRH is false for the $L$-function of some imaginary quadratic Dirichlet character. See Ireland and Rosen's number theory book, p. 359. Jul 11, 2013 at 22:31
• Before the work of Heilbronn, Mordell had shown that if infinitely many imaginary quadratic fields have the same class number (any common value) then RH for the Riemann zeta-function is true. Thus if RH is false, any positive integer can be the class number of finitely many imaginary quadratic fields. Jul 11, 2013 at 22:33
• About 10,000 published papers would be rendered vacuous. Jul 11, 2013 at 23:42
• @GerryMyerson Let's not throw the baby out with the bath water! Jul 30 at 5:02

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
• Such results "for a range of $d$" already exist, in terms of zeros which are closer than average (see Stark's thesis, or Montgomery/Weinberger I think though they use low-zeros of Dirichlet $L$-functions, same principle). So does this really have anything to do with RH failing per se? eudml.org/doc/205246mathoverflow.net/questions/55959/… Jul 12, 2013 at 1:28
• @v08ltu:Yes, I have Stark's thesis in front of me. Chapter 2 is the contrapositive, for class number 1. A 10th such discriminant $-p$ would imply the Riemann Hypothesis up to height $\sqrt{p}/2$. My answer above does say 'and others.' Jul 12, 2013 at 15:33