# Refinements of the Riemann hypothesis

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?

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What Hugh Montgomery has to say at the end of the email math.niu.edu/~rusin/known-math/99/primedistr might interest you. –  KConrad Oct 20 '10 at 17:53

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.

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You should also mention that the uncertainty principal for the Fourier transform shows that even if we'd have a description of the constants on one side of the explicit formula, the finer structure of the constants on the other side remains hidden. So although the Riemann zeros seems to know everything about the primes, their exact positions will not tell everything about them just by applying the explicit formula a la Weil. –  Marc Palm Apr 25 '11 at 17:12

For the Riemann zeta function, there are several conjectures that are stronger than the Riemann Hypothesis and these imply stronger results on distribution of prime numbers than what are known on RH alone. The most notable is probably the Pair Correlation Conjecture of Montgomery. This is a conjecture that assumes the Riemann hypothesis and predicts on finer distribution of the gaps between the imaginary parts of the zeros. There are similar conjectures due to Katz and Sarnak for many other $L$-functions. A good place to read about this is the article by Katz and Sarnak http://www.ams.org/journals/bull/1999-36-01/S0273-0979-99-00766-1/home.html

About the conjecture you have mentioned, I don't know if any standard conjecture on the zeros of the Riemann zeta function implies it. Heath-Brown has a result on the application of PCC on gaps between consecutive primes but I don't think it comes anywhere near this one.

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Do you have a reference for Heath-Brown's result? –  wood Oct 20 '10 at 21:40
–  Idoneal Oct 21 '10 at 9:07

The paper arxiv:1008.2381 has a claim for the prime gap

p_(n+1) - p_n = O(log^2(p_n)/loglog(p_n)),

assuming the Riemann hypothesis.

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Looks screwy to me. Author uses nothing new to prove result better than what's been long known to be best possible. Formula (7) looks suspicious, what with a main term we expect to be bounded and an error term $o(n/\log n)$. Disclaimer - I'm not an expert on zeta, though I have played one in the classroom. –  Gerry Myerson Oct 30 '10 at 23:05
It is believed that infinitely often $p_{n+1}-p_n \gg log^2 p_n$, so the result claimed in the arXiv paper seems very unlikely to be true. See page 5 of this: ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/… –  Micah Milinovich Apr 16 '11 at 19:22
In that paper equation (10) is the following: [d_n+(o(p_{n+1})-o(p_n))<c\log^2 n/\log\log n] where $d_n=p_{n+1}-p_n$. It is absolutely impossible that from this he gets anything better than $d_n=o(p_n)$, which is nothing. Further, on pp 7-8, he breaks into 4 cases, all stating that something occurs for all $n\geq n_0$. These cases do not cover all possibilities. –  Péter Komjáth Sep 23 '11 at 14:16