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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28$\begingroup$ 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. $\endgroup$– user9072Commented Jul 11, 2013 at 15:11
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8$\begingroup$ 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. $\endgroup$– KConradCommented Jul 11, 2013 at 22:31
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3$\begingroup$ 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. $\endgroup$– KConradCommented Jul 11, 2013 at 22:33
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47$\begingroup$ About 10,000 published papers would be rendered vacuous. $\endgroup$– Gerry MyersonCommented Jul 11, 2013 at 23:42
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1$\begingroup$ @GerryMyerson Let's not throw the baby out with the bath water! $\endgroup$– Jay OrdwayCommented Jul 30, 2023 at 5:02
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1 Answer
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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 'Deuring-Heilbronn phenomenon,' with results due to these two and others beginning in the 1930's. For an elementary account, see
https://arxiv.org/abs/1201.0713 (Jeffrey Stopple: Elementary Deuring-Heilbronn Phenomenon)
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1$\begingroup$ 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/205246 mathoverflow.net/questions/55959/… $\endgroup$– v08ltuCommented Jul 12, 2013 at 1:28
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3$\begingroup$ @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.' $\endgroup$– StoppleCommented Jul 12, 2013 at 15:33