Consequences of the negation of the Riemann hypothesis

Question

There are many sources documenting the consequences of the Riemann hypothesis, but I can't find one discussing the consequences of its negation, particularly concerning the prime distribution.

Can anyone refer some literature regarding this topic?

Answer 1

It seems that there are not too many articles discussing the consequences of the failure of RH. More often one can read that the failure would be a "disaster". Concerning the prime distribution, Enrico Bombieri puts it as follows: “The failure of the Riemann Hypothesis would create havoc in the distribution of prime numbers”.

If the Riemann hypothesis were false then also $$ \pi(x)=\int_2^x \frac{dt}{\log(t)}+O(\sqrt{x}\log(x)) $$ were false, i.e., the error term would be worse. In this case the question would be how good (how large) the zero-free regions of $\zeta(s)$ really are. Certainly the prime distribution then would have a very interesting behaviour.
On the other hand, De la Vallee-Poussin already constructed in 1896 a good zero-free region for $\zeta(s)$ (good enough to prove PNT at least) yielding an error term $O(xe^{-c\log(x)})$.