This is the seventh 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.

The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to . Very recently, we have been able to exploit GEH more fully, leading to a promising new expansion of the sieve support region. The problem now reduces to the following:

Problem 1 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

when ;

when ; when ;

when ; when ; and such that we have the inequality

An affirmative answer to this question will imply on GEH. We are “within two percent” of this claim; we cannot quite reach yet, but have got as far as . However, we have not yet fully optimised in the above problem. In particular, the simplex

is now available, and should lead to some noticeable improvement in the numerology.

There is also a very slim chance that the twin prime conjecture is now provable on GEH. It would require an affirmative solution to the following problem:

Problem 2 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

when ;

when ; when ; and such that we have the inequality

We suspect that the answer to this question is negative, but have not formally ruled it out yet.

For the rest of this post, I will justify why positive answers to these sorts of variational problems are sufficient to get bounds on (or more generally ).

— 1. Crude sieve bounds —

Let the notation be as in the Polymath8a paper, thus we have an admissible tuple , a residue class with coprime to for all , and an asymptotic parameter going off to infinity. It will be convenient to use the notation

We let be the interval

and for each fixed smooth compactly supported function , we let denote the divisor sum

We wish to understand the correlation of various products of divisor sums on . For instance, in this previous blog post, the asymptotic

was established whenever one has the support condition

where is the outer edge of the support of , and

We are now interested in understanding the asymptotics when (2) fails. We have a crude pointwise upper bound:

Lemma 3 Let be a fixed smooth compactly supported function. Then for any natural number , for any fixed . More generally, for any fixed number of fixed smooth compactly supported functions, one has

Proof: We extend smoothly to all of as a compactly supported function, and write the Fourier expansion

for some rapidly decreasing function . Then

Taking absolute values, we conclude that

Since , the first claim now follows from the rapid decrease of . To prove the second claim, we use the first claim to bound the left-hand side of (3) by

Bounding

and

the claim follows after a change of variables.

Lemma 4 For each , let and be fixed, and let be fixed smooth compactly supported functions. Then We also have the variant for any and , where is the least prime factor of .

The intuition here is that each of the is mostly bounded and mostly supported on the for which is almost prime (so in particular is bounded), which has a density of about in .

Proof: From (3) (and bounding ), we can bound the left-hand side of (4) by

Let be a small fixed number ( will do). For each , we let be the prime factors of in increasing order (counting multiplicity), and let be the largest product of consecutive primes factors that is bounded by . In particular, we see that

and hence

which in particular implies that . This implies that

Now observe that , where is -rough (i.e. no prime factors less than ). In particular, it is -rough. Thus we can bound the left-hand side of (4) by

By using a standard upper bound sieve (and taking small enough), the quantity

may be bounded by

Since , we can thus bound the left-hand side of (4) by

We can bound this by

(strictly speaking one has some additional contribution coming from repeated primes , but these can be eliminated in a number of ways, e.g. by restricting initially to square-free ). By Mertens’ theorem we have

\endand then by summing the series in , we can bound the left-hand side of (4) by

which for large enough is as required. This proves (4).

The proof of (5) is similar, except that (assuming small, as we may) is forced to be at least , and is at most . From this we may effectively extract an additional factor of (times a loss of due to having to reduce to ), which gives rise to the additional gain of .

— 2. The generalised Elliott-Halberstam conjecture —

We begin by stating the conjecture more formally, using (a slightly weaker form of) the version from this paper of Bombieri, Friedlander, and Iwaniec. We use the notation from the Polymath8a paper.

Conjecture 5 (GEH) Let for some fixed , be such that , and let be coefficient sequences at scale . Then for any fixed .

We use GEH to refer to the assertion that holds for all . As shown by Motohashi, a modification of the proof of the Bombieri-Vinogradov theorem shows that is true for . (It is possible that some modification of the arguments of Zhang give some weak version of GEH for some slightly above , but we will not focus on that topic here.)

For our purposes, we will need to apply GEH to functions supported on products of primes for a fixed (generalising the von Mangoldt function, which is the focus of the Elliott-Halberstam conjecture EH). More precisely, we have

Proposition 6 Assume holds. Let and be fixed, let , and let be a fixed smooth function. Let be the function defined by setting whenever is the product of distinct primes with for some fixed , and otherwise. Then for any fixed .

Remark: it may be possible to get some version of this proposition just from EH using Bombieri’s asymptotic sieve.

Proof: (Sketch) This is a standard partitioning argument (not sure where it appears first, though). We choose a fixed that is sufficiently large depending on . We can decompose the primes from to into intervals . This splits into pieces, depending on which intervals the lie in. The contribution when two primes lie in the same interval, or when the products of the specified intervals touches the boundary of , can be shown to be negligible by crude divisor function estimates if is large enough (a similar argument appears in the Polymath8a paper), basically because there are only such terms, and each one contributes to the total. For the remaining pieces, one can approximate by a constant, up to errors which can also be shown to be negligible by crude estimates for large enough (each term contributes ), and then can be modeled by a convolution of coefficient sequences at various scales between and , at which point one can use GEH to conclude.

Corollary 7 Assume holds for some . Let and be fixed, let be fixed and smooth, and let be as in the previous proposition. Let be a divisor sum of the form where are coefficients supported on the range . Then for any fixed . Similarly for permutations of the .

Proof: (Sketch) We can rearrange as

Using the previous proposition, and the Chinese remainder theorem, we may approximate by

plus negligible errors (here we need the crude bounds on and some standard bounds on the divisor function), thus

A similar argument gives

and the claim follows by combining the two assertions.

Next, from Mertens’ theorem one easily verifies that

where is the expected density of primes in , and the measure on is the one induced from Lebesgue measure on the first coordinates . (One could improve the term to here by using the prime number theorem, but it isn’t necessary for our analysis.)

Applying (1), we thus have

Corollary 8 Assume holds for some . Let and be fixed, let be fixed and smooth, and let be as in the previous proposition. For , let be smooth compactly supported functions with . Then where and for .

— 3. Some integration identities —

Lemma 9 Let be a smooth function. Then where .

Proof: Making the change of variables , the integral can be written as

which by symmetry is equal to

which after expanding out the square and using symmetry is equal to

and the claim follows from the fundamental theorem of calculus.

Iterating this lemma times, we conclude that

for any , where the first integral is integrated using . In particular, discarding the final term (which is non-negative) and then letting , we obtain the inequality

In fact we have equality:

Proposition 10 Let be smooth. Then In particular, by depolarisation we have for smooth .

Proof: Let be a small quantity, and write

From Lemma 9 we have

where

From another application of Lemma 9 we have

where

Iterating this, we see that

for any , where

If , then vanishes, thus

For , we may rewrite

where

and . By Fubini’s theorem, we have

Discarding the constraints and using and , we conclude that

Summing over using (6), we see that

since by the smoothness of , we conclude that

and thus by (9)

Direct computation also shows that , hence

and thus

But by the monotone convergence theorem, as , converges to

Thus we can complement (6) with the matching upper bound, giving the claim.

We can rewrite the above identity using the following cute identity (which presumably has a name?)

Lemma 11 For any positive reals with , one has where ranges over the permutations of .

Thus for instance

and so forth.

Proof: We induct on . The case is trivial. If and the claim has already been proven for , then from induction hypothesis one has

for each . Summing over , we obtain the claim.

Proposition 12 Let be smooth. Then

Proof: Average (7) over permutations of the and use Lemma 11.

This gives us a variant of

Corollary 13 Assume holds for some . For , let be smooth compactly supported functions with . Then where for .

Proof: Let . By (5), we have

so by paying a cost of , we may restrict to which are -rough, and are thus of the form for some and . For (restricting to squarefree integers to avoid technicalities), we have

and similarly for . Using this and Corollary 13, we may write the left-hand side of (10) as

Sending and using dominated convergence and Proposition 11, we obtain the claim.

Taking linear combinations, we conclude the usual “denominator” asymptotic

with

whenever is supported on a polytope (not necessarily convex) with

and is a finite linear combination of tensor products of smooth compactly supported functions. We use this as a replacement for the denominator estimate in this previous blog post, we obtain the criteria described above.