This is the fifth thread for the Polymath8b project to obtain new bounds for the quantity
either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for has been shaved a little from to , and we have very recently achieved the bound on the generalised Elliott-Halberstam conjecture GEH, formulated as Conjecture 1 of this paper of Bombieri, Friedlander, and Iwaniec. We also have explicit bounds for for , both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for as .
The basic strategy for bounding still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible -tuple , one needs to locate a non-negative sieve weight , supported on an interval for a large , such that the ratio
is asymptotically larger than as ; this will show that . Thus one wants to locate a sieve weight for which one has good lower bounds on the numerator and good upper bounds on the denominator.
One can modify this paradigm slightly, for instance by adding the additional term to the numerator, or by subtracting the term from the numerator (which allows one to reduce the bound to ); however, the numerical impact of these tweaks have proven to be negligible thus far.
Despite a number of experiments with other sieves, we are still relying primarily on the Selberg sieve
where is the divisor sum
with , is the level of distribution ( if relying on Bombieri-Vinogradov, if assuming Elliott-Halberstam, and (in principle) if using Polymath8a technology), and is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions one is permitted to use.
The baseline bounds for the numerator and denominator in (1) (as established for instance in this previous post) are as follows. If is supported on the simplex
and we define the mixed partial derivative by
then the denominator in (1) is
Similarly, the numerator of (1) is
Thus, if we let be the supremum of the ratio
whenever is supported on and is non-vanishing, then one can prove whenever
We can improve this baseline in a number of ways. Firstly, with regards to the denominator in (1), if one upgrades the Elliott-Halberstam hypothesis to the generalised Elliott-Halberstam hypothesis (currently known for , thanks to Motohashi, but conjectured for ), the asymptotic (2) holds under the more general hypothesis that is supported in a polytope , as long as obeys the inclusion
examples of polytopes obeying this constraint include the modified simplex
the dilated simplex
and the truncated simplex
See this previous post for a proof of these claims.
With regards to the numerator, the asymptotic (3) is valid whenever, for each , the marginals vanish outside of . This is automatic if is supported on , or on the slightly larger region , but is an additional constraint when is supported on one of the other polytopes mentioned above.
More recently, we have obtained a more flexible version of the above asymptotic: if the marginals vanish outside of for some , then the numerator of (1) has a lower bound of
A proof is given here. Putting all this together, we can conclude
Theorem 1 Suppose we can find and a function supported on a polytope obeying (4), not identically zero and with all marginals vanishing outside of , and with
Then implies .
In principle, this very flexible criterion for upper bounding should lead to better bounds than before, and in particular we have now established on GEH.
Another promising direction is to try to improve the analysis at medium (more specifically, in the regime ), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using and using the baseline analysis, basically for two reasons:
- We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex in medium dimension.
- The estimates produced by Polymath8a involve a parameter, which introduces additional restrictions on the support of (conservatively, it restricts to where and ; it should be possible to be looser than this (as was done in Polymath8a) but this has not been fully explored yet). This then triggers the previous obstacle of having to integrate on something other than a simplex.
However, these look like solvable problems, and so I would expect that further unconditional improvement for should be possible.