I have often read that the Riemann hypothesis is somewhat a statement like:
The primes are as regularly distributed as we can hope for.
For example $\pi(x) = Li(x)+ O(x^{\sigma+\epsilon})$ for any $\epsilon>0$ as long as there are no zeros of $\zeta$ for any
$s \in \mathbb{C}$ with $\Re s > \sigma$. And of course $\sigma=\frac{1}{2}$ is the best we can hope for.
However there are instances/problems where the Riemann hypothesis does not give the "right conjectural"
answers about the distribution of primes.
Let me state one example I recently read about.
Cramer's conjecture http://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture
asserts that
$$ p_{n+1} – p_n = O(\log^2 p_n ). $$
Here, the Riemann hypothesis only gives $p_{n+1} – p_n = O(\sqrt{p_n} \log p_n )$. So, in some sense RH does not give the best we can hope for.
I am now asking for for refinements that could achieve this. Could, for example, Cramer's conjecture be deduced from a refinement of $\pi(x) = Li(x)+ O(x^{\frac{1}{2}+\epsilon})$ (under RH), maybe making the O term more precise. And how would this be reflected in terms of the $\zeta$-function and its zeros.
I know there are generalizations to other $L$-functions and refinements like the pair correlation conjecture or predictions from random matrix theory(though I do not have any clue about this). But I do not know whether these
can help to resolve for example Cramer's conjecture. My question is somewhat unprecise (if anyone can write it up better, feel free to edit.)
: Is there a "Super Riemann Hypothesis" predicting stronger properties of the $\zeta$ or other $L$-functions that would settle most questions about the distribution of primes?
EDIT: Thank you for the answers so far. Since Charles pointed out that my question is really to imprecise as we could prove anything about the primes if we knew the zeros of $\zeta$ I am going to rephrase my question inspired by the interesting article of Heath-Brown mentioned by Idoneal:
What sort of properties do we have to know/assume about the zeros of $\zeta$ in order to deduce Cramer's conjecture?
Best Answer
Yes, your question is imprecise. If we knew exactly where the zeta zeroes were, we could answer any question about the primes that could be formulated by means of the explicit formulae. In crude terms, the primes are obtained from the zeroes by an integral transform. Specific questions can depend, for example, on rational dependencies between the imaginary parts of the zeroes. There is quite a large literature on conditional results on primes, and if you are asking whether there is a way of packaging it all up neatly I'd expect the more reasonable answer to be "no" rather than "yes". Several generations of analytic number theorists have contributed.