Zeta functions versus Cramer's conjecture

Question

A mathematics professor today asked me if Cramer's conjecture on prime gaps has anything to do with Riemann Zeta function. I did not know but my guess was somehow Cramer's conjecture captures local properties while zeta functions tell global picture (from answer in here https://math.stackexchange.com/questions/1087381/cramer-and-riemann-conjecture-implication). Is there a connection between Cramer's conjecture-zeta functions? Cramer proved a weak gap assuming Riemann Hypothesis. Is there any other information from zeta function that can be used to address prime gaps at all (apart from information needed for GRH)?

Answer 1

It was shown by Heath-Brown that a suitable form of the pair correlation conjecture of Montgomery, in conjunction with RH, could improve Cramer's bound $p_{n+1}-p_n \ll p_n^{1/2} \log p_n$ slightly to $p_{n+1}-p_n \ll p_n^{1/2} \log^{1/2} p_n$. This appears to be close to the limit of what one can do purely from local information on the zeta function. Of course, thanks to the explicit formula, one could in principle extract more precise information on $p_{n+1}-p_n$ in terms of global information about zeta (e.g. involving cancellation between terms from distant zeroes) but nobody expects that such information could be at all accessible from any of our known methods, other than through the circular device of somehow first establishing bounds on $p_{n+1}-p_n$ by other, non-zeta function-based, methods, and then transforming this to an assertion about the zeta function.

Local information on zeta, such as RH or pair correlation, is more effective when dealing with averaged prime gaps, such as the second moment of $p_{n+1}-p_n$, for instance under the same hypotheses as before, Heath-Brown was able to show that $(p_{n+1}-p_n)^2$ was bounded by $O(\log^3 p_n)$ on the average.