This is the second 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 are:
- (Maynard) .
- (Polymath8b, tentative) .
- (Polymath8b, tentative) for sufficiently large .
- (Maynard) Assuming the Elliott-Halberstam conjecture, , , and .
Following the strategy of Maynard, the bounds on proceed by combining four ingredients:
- Distribution estimates or for the primes (or related objects);
- Bounds for the minimal diameter of an admissible -tuple;
- Lower bounds for the optimal value to a certain variational problem;
- Sieve-theoretic arguments to convert the previous three ingredients into a bound on .
Accordingly, the most natural routes to improve the bounds on are to improve one or more of the above four ingredients.
Ingredient 1 was studied intensively in Polymath8a. The following results are known or conjectured (see the Polymath8a paper for notation and proofs):
- (Bombieri-Vinogradov) is true for all .
- (Polymath8a) is true for .
- (Polymath8a, tentative) is true for .
- (Elliott-Halberstam conjecture) is true for all .
Ingredient 2 was also studied intensively in Polymath8a, and is more or less a solved problem for the values of of interest (with exact values of for , and quite good upper bounds for for , available at this page). So the main focus currently is on improving Ingredients 3 and 4.
For Ingredient 3, the basic variational problem is to understand the quantity
for bounded measurable functions, not identically zero, on the simplex
with being the quadratic forms
Equivalently, one has
where is the positive semi-definite bounded self-adjoint operator
so is the operator norm of . Another interpretation of is that the probability that a rook moving randomly in the unit cube stays in simplex for moves is asymptotically .
We now have a fairly good asymptotic understanding of , with the bounds
holding for sufficiently large . There is however still room to tighten the bounds on for small ; I’ll summarise some of the ideas discussed so far below the fold.
For Ingredient 4, the basic tool is this:
Thus, for instance, it is known that and , and this together with the Bombieri-Vinogradov inequality gives . This result is proven in Maynard’s paper and an alternate proof is also given in the previous blog post.
We have a number of ways to relax the hypotheses of this result, which we also summarise below the fold.
— 1. Improved sieving —
A direct modification of the proof of Theorem 1 also shows:
Here is defined for the truncated simplex in the obvious fashion. This allows us to use the MPZ-type bounds obtained in Polymath8a, at the cost of requiring the test functions to have somewhat truncated support. Fortunately, in the large setting, the functions we were using had such a truncated support anyway. It looks likely that we can replace the cube by significantly larger regions by using the (multiple) dense divisibility versions of , but we have not yet looked into this.
It also appears that if one generalises the Elliott-Halberstam conjecture to also encompass more general Dirichlet convolutions than the von Mangoldt function (see e.g. Conjecture 1 of Bombieri-Friedlander-Iwaniec), then one can enlarge the simplex in Theorem 1 (and probably for Theorem 2 also) to the slightly larger region
Basically, the reason for this is that the restriction to the simplex (as opposed to ) is only needed to control the sum , but by splitting into products of simpler divisor sums, and using the Elliott-Halberstam hypothesis to control one of the factors, it looks like one can still control error terms in the larger region (but this will have to be checked at some point, if we end up using this refinement). This is only likely to give a slight improvement, except when is small; from the inclusions
and a scaling argument we see that
Assume EH. To improve the bound to , it suffices to obtain a bound of the form
With given in terms of a cutoff function , the left-hand side can be computed as usual as
while we have the upper bound
and other bounds may be possible. (This is discussed in this comment.)
For higher , it appears that similar maneuvers will have a relatively modest impact, perhaps shaving or so off of the current values of .
— 2. Upper bound on —
We have the upper bound
for any . To see this, observe from Cauchy-Schwarz that
The final factor is , and so
Summing in and noting that on the simplex we have
and the claim follows.
Setting , we conclude that
for sufficiently large . There may be some room to improve these bounds a bit further.
— 3. Lower bounds on —
For small , one can optimise the quadratic form
by specialising to a finite-dimensional space and then performing the appropriate linear algebra. It is known that we may restrict without loss of generality to symmetric ; one could in principle also restrict to the functions of the form
for some symmetric function (indeed, morally at least should be an eigenfunction of ), although we have not been able to take much advantage of this yet.
For large , we can use the bounds
for any and any with ; we can also start with a given and improve it by replacing it with (normalising in if desired), and perhaps even iterating and accelerating this process.
The basic functions we have been using take the form
Then , and
where are iid random variables on with density . By Chebyshev’s inequality we then have
if , where
A lengthier computation for gives