Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only known for degree 3 by work of Wooley. See this survey of Wooley for a more discussion.

It seems to be folklore that a proof of this conjecture *should* imply improved bounds on the Riemann zeta function / an improved zero free region for the zeta function / an improved error term on the prime number theorem (among other things, such as progress on Waring's problem). These, of course, are some of the most highly prized problems in analytic number theory and have been stuck for decades.

That said, to the best of my knowledge there is no blackbox reduction for these applications and one likely also needs more explicit information about the dependence of constants on various parameters in these results to reach these applications.

What is the potential of these methods? In other words, what are the implications of the most optimistic dependencies in the mean value theorem (or possible generalizations)?

potentialof a certain new approach is usually hard to answer ... . $\endgroup$ – Stefan Kohl Dec 9 '15 at 9:312more comments