This is the fourth thread for the Polymath8b project to obtain new bounds for the quantity

$\displaystyle H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),$

either for small values of ${m}$ (in particular ${m=1,2}$) or asymptotically as ${m \rightarrow \infty}$. The previous thread may be found here. The currently best known bounds on ${H_m}$ are:

• (Maynard) Assuming the Elliott-Halberstam conjecture, ${H_1 \leq 12}$.
• (Polymath8b, tentative) ${H_1 \leq 272}$. Assuming Elliott-Halberstam, ${H_2 \leq 272}$.
• (Polymath8b, tentative) ${H_2 \leq 429{,}822}$. Assuming Elliott-Halberstam, ${H_4 \leq 493{,}408}$.
• (Polymath8b, tentative) ${H_3 \leq 26{,}682{,}014}$. (Presumably a comparable bound also holds for ${H_6}$ on Elliott-Halberstam, but this has not been computed.)
• (Polymath8b) ${H_m \leq \exp( 3.817 m )}$ for sufficiently large ${m}$. Assuming Elliott-Halberstam, ${H_m \ll m e^{2m}}$ for sufficiently large ${m}$.

While the ${H_1}$ bound on the Elliott-Halberstam conjecture has not improved since the start of the Polymath8b project, there is reason to hope that it will soon fall, hopefully to ${8}$. This is because we have begun to exploit more fully the fact that when using “multidimensional Selberg-GPY” sieves of the form

$\displaystyle \nu(n) := \sigma_{f,k}(n)^2$

with

$\displaystyle \sigma_{f,k}(n) := \sum_{d_1|n+h_1,\dots,d_k|n+h_k} \mu(d_1) \dots \mu(d_k) f( \frac{\log d_1}{\log R},\dots,\frac{\log d_k}{\log R}),$

where ${R := x^{\theta/2}}$, it is not necessary for the smooth function ${f: [0,+\infty)^k \rightarrow {\bf R}}$ to be supported on the simplex

$\displaystyle {\cal R}_k := \{ (t_1,\dots,t_k)\in [0,1]^k: t_1+\dots+t_k \leq 1\},$

but can in fact be allowed to range on larger sets. First of all, ${f}$ may instead be supported on the slightly larger polytope

$\displaystyle {\cal R}'_k := \{ (t_1,\dots,t_k)\in [0,1]^k: t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 1$

$\displaystyle \hbox{ for all } j=1,\dots,k\}.$

However, it turns out that more is true: given a sufficiently general version of the Elliott-Halberstam conjecture ${EH[\theta]}$ at the given value of ${\theta}$, one may work with functions ${f}$ supported on more general domains ${R}$, so long as the sumset ${R+R := \{ t+t': t,t'\in R\}}$ is contained in the non-convex region

$\displaystyle \bigcup_{j=1}^k \{ (t_1,\dots,t_k)\in [0,\frac{2}{\theta}]^k: t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 2 \} \cup \frac{2}{\theta} \cdot {\cal R}_k, \ \ \ \ \ (1)$

and also provided that the restriction

$\displaystyle (t_1,\dots,t_{j-1},t_{j+1},\dots,t_k) \mapsto f(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k) \ \ \ \ \ (2)$

is supported on the simplex

$\displaystyle {\cal R}_{k-1} := \{ (t_1,\dots,t_{j-1},t_{j+1},\dots,t_k)\in [0,1]^{k-1}:$

$\displaystyle t_1+\dots+t_{j-1}+t_{j+1}+\dots t_k \leq 1\}.$

More precisely, if ${f}$ is a smooth function, not identically zero, with the above properties for some ${R}$, and the ratio

$\displaystyle \sum_{j=1}^k \int_{{\cal R}_{k-1}} f_{1,\dots,j-1,j+1,\dots,k}(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k)^2 \ \ \ \ \ (3)$

$\displaystyle dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k$

$\displaystyle / \int_R f_{1,\dots,k}^2(t_1,\dots,t_k)\ dt_1 \dots dt_k$

is larger than ${\frac{2m}{\theta}}$, then the claim ${DHL[k,m+1]}$ holds (assuming ${EH[\theta]}$), and in particular ${H_m \leq H(k)}$.

I’ll explain why one can do this below the fold. Taking this for granted, we can rewrite this criterion in terms of the mixed derivative ${F := f_{1,\dots,k}}$, the upshot being that if one can find a smooth function ${F}$ supported on ${R}$ that obeys the vanishing marginal conditions

$\displaystyle \int F( t_1,\dots,t_k )\ dt_j = 0$

whenever ${1 \leq j \leq k}$ and ${t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k > 1}$, and the ratio

$\displaystyle \frac{\sum_{j=1}^k J_k^{(j)}(F)}{I_k(F)} \ \ \ \ \ (4)$

is larger than ${\frac{2m}{\theta}}$, where

$\displaystyle I_k(F) := \int_R F(t_1,\dots,t_k)^2\ dt_1 \dots dt_k$

and

$\displaystyle J_k^{(j)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1/\theta} F(t_1,\dots,t_k)\ dt_j)^2 dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k$

then ${DHL[k,m+1]}$ holds. (To equate these two formulations, it is convenient to assume that ${R}$ is a downset, in the sense that whenever ${(t_1,\dots,t_k) \in R}$, the entire box ${[0,t_1] \times \dots \times [0,t_k]}$ lie in ${R}$, but one can easily enlarge ${R}$ to be a downset without destroying the containment of ${R+R}$ in the non-convex region (1).) One initially requires ${F}$ to be smooth, but a limiting argument allows one to relax to bounded measurable ${F}$. (To approximate a rough ${F}$ by a smooth ${F}$ while retaining the required moment conditions, one can first apply a slight dilation and translation so that the marginals of ${F}$ are supported on a slightly smaller version of the simplex ${{\cal R}_{k-1}}$, and then convolve by a smooth approximation to the identity to make ${F}$ smooth, while keeping the marginals supported on ${{\cal R}_{k-1}}$.)

We are now exploring various choices of ${R}$ to work with, including the prism

$\displaystyle \{ (t_1,\dots,t_k) \in [0,1/\theta]^k: t_1+\dots+t_{k-1} \leq 1 \}$

and the symmetric region

$\displaystyle \{ (t_1,\dots,t_k) \in [0,1/\theta]^k: t_1+\dots+t_k \leq \frac{k}{k-1} \}.$

By suitably subdividing these regions into polytopes, and working with piecewise polynomial functions ${F}$ that are polynomial of a specified degree on each subpolytope, one can phrase the problem of optimising (4) as a quadratic program, which we have managed to work with for ${k=3}$. Extending this program to ${k=4}$, there is a decent chance that we will be able to obtain ${DHL[4,2]}$ on EH.

We have also been able to numerically optimise ${M_k}$ quite accurately for medium values of ${k}$ (e.g. ${k \sim 50}$), which has led to improved values of ${H_1}$ without EH. For large ${k}$, we now also have the asymptotic ${M_k=\log k - O(1)}$ with explicit error terms (details here) which have allowed us to slightly improve the ${m=2}$ numerology, and also to get explicit ${m=3}$ numerology for the first time.

— 1. More details on the relaxed support conditions —

Fix ${R}$ as above, which we take to be a polytope that is also a downset, and let ${{\cal F}}$ be the class of smooth functions ${f}$, not identically zero, supported on ${R}$ whose restrictions (2) are all supported on ${{\cal R}_{k-1}}$. We need a technical analysis result, that any ${f}$ in ${{\cal F}}$ can be approximated by a finite linear combination ${\tilde f}$ of tensor products ${f_1(t_1) \dots f_k(t_k)}$, each of which supported in a cube in the interior of ${R}$ of arbitrarily small specified side length, such that the ratio (3) for ${f}$ and ${\tilde f}$ are arbitrary close. This is almost a standard application of Stone-Weierstrass, but the vanishing properties of ${f}$ mean that some care has to be taken.

First, by using a slight dilation to shrink ${f}$ a bit, we can assume that ${f}$ is supported on ${(1-\epsilon) \cdot R}$ for some small ${\epsilon>0}$, and all the restrictions (2) supported on ${(1-\epsilon)\cdot {\cal R}_{k-1}}$, while only modifying (4) by an epsilon. We extend ${f}$ from ${[0,\infty)^k}$ to ${{\bf R}^k}$ by setting

$\displaystyle f(t_1,\dots,t_k):= f(\max(t_1,0),\dots,\max(t_k,0));$

${f}$ is now Lipschitz and piecewise smooth instead of smooth. Applying a translation (and then truncating back to ${[0,+\infty)^k}$, we can keep ${f}$ supported in ${(1-\frac{\epsilon}{2}) \cdot R}$ while also vanishing in a neighbourhood of the regions

$\displaystyle \{ (t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k): \sum_{i \neq j} t_i \geq 1 \} \ \ \ \ \ (5)$

for each ${j}$. This does not affect (3), but ${f}$ is now piecewise smooth and Lipschitz rather than smooth. However, by convolving with an appropriate approximation to the identity, we can make ${f}$ smooth again, while still supported in a slightly smaller version of ${R}$ obeying the above-mentioned vanishing property. By using the Lebesgue differentiation theorem one can show that (3) is almost unaffected by this if we make the approximation to the identity sufficiently narrow. We can then smoothly partition ${f}$ into smooth functions supported on small cubes, each contained in ${R}$ and avoiding the regions (5) even after a slight dilation, and then by using Fourier series one can approximate each such component of ${f}$ by a finite combination of tensor products of smooth functions, that are still supported in ${R}$ and avoiding (5), and the claim follows.

From all this (and the fact that the property of (3) being larger than ${\frac{2m}{\theta}}$ is an open condition) we may assume that ${f}$ is a finite sum of tensor products ${f_i}$ supported on small cubes in ${R}$ avoiding (5). For the GPY argument to work (as laid out in this previous post), we need the asymptotics

$\displaystyle \sum_{x\leq n \leq 2x: n = b\ (W)} \nu(n) = (\alpha_{f,f}+o(1)) B\frac{x}{W}$

and

$\displaystyle \sum_{x\leq n \leq 2x: n = b\ (W)} \nu(n) \theta(n+h_j) = (\beta_{j,f,f}+o(1)) B\frac{x}{W} \log R$

to hold, where

$\displaystyle B := (\frac{W}{\phi(W)})^k \frac{1}{\log^k R}$

$\displaystyle \alpha_{f,f} = \int_{{\cal R}_k} f_{1,\dots,k}^2(t_1,\dots,t_k)\ dt_1 \dots dt_k$

$\displaystyle \beta_{j,f,f} := \int_{{\cal R}_{k-1}} f_{1,\dots,j-1,j+1,\dots,k}^2(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k)$

$\displaystyle dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k.$

We can generalise these bounds slightly to the depolarised bounds

$\displaystyle \sum_{x\leq n \leq 2x: n = b\ (W)} \sigma_{f,k}(n) \sigma_{f',k}(n) = (\alpha_{f,f'}+o(1)) B\frac{x}{W} \ \ \ \ \ (6)$

and

$\displaystyle \sum_{x\leq n \leq 2x: n = b\ (W)} \sigma_{f,k}(n) \sigma_{f',k}(n) \theta(n+h_j) = (\beta_{j,f,f'}+o(1)) B\frac{x}{W} \log R \ \ \ \ \ (7)$

to hold, where

$\displaystyle \alpha_{f,f'} = \int_{{\cal R}_k} f_{1,\dots,k}(t_1,\dots,t_k) f'_{1,\dots,k}(t_1,\dots,t_k)\ dt_1 \dots dt_k$

$\displaystyle \beta_{j,f,f'} := \int_{{\cal R}_{k-1}} f_{1,\dots,j-1,j+1,\dots,k}(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k)$

$\displaystyle f'_{1,\dots,j-1,j+1,\dots,k}(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k)\ dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k.$

and ${f,f'}$ are linear combinations of smooth tensor products supported on small cubes in ${R}$. By linearity, we may assume that ${f,f'}$ are each supported on a small cube ${Q,Q'}$ respectively.

For the sums (7), the “${\beta}$” calculations in the previous blog post already suffice, because ${Q+Q'}$ avoids the (5) and so divisors appearing in ${\sigma_f(n) \sigma_{f'}(n)}$ have magnitude at most ${R}$ (or ${x^{-\epsilon} R}$ for some small ${\epsilon>0}$). For (6), a compactness argument and the inclusion of ${R+R}$ in (1) shows that if the cubes are small enough, we have either

$\displaystyle Q+Q' \subset (\frac{2}{\theta}-\epsilon) \cdot {\cal R}_k$

or

$\displaystyle Q+Q' \subset \{ (t_1,\dots,t_k)\in [0,\frac{2}{\theta}-\epsilon]^k:$

$\displaystyle t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 2-\epsilon \}$

for some ${j}$ and some ${\epsilon>0}$. In the former case, we can compute much as in the “${\alpha}$” calculation in the previous blog post. Now suppose that we are in the latter case. Without loss of generality let us take ${j=k}$, thus

$\displaystyle Q+Q' \subset \{ (t_1,\dots,t_k)\in [0,\frac{2}{\theta}-\epsilon]^k: t_1+\dots+t_{k-1} \leq 2-\epsilon \}.$

If we use the tensor product structure to write

$\displaystyle f(t_1,\dots,t_{k}) = \tilde f(t_1,\dots,t_{k-1}) f_k(t_k)$

and

$\displaystyle f'(t_1,\dots,t_{k}) = \tilde f'(t_1,\dots,t_{k-1}) f'_k(t_k)$

for smooth ${\tilde f, f_k, \tilde f', f'_k}$ supported in appropriate cubes or intervals, we can refactor

$\displaystyle \sum_{x\leq n \leq 2x: n = b\ (W)} \sigma_f(n) \sigma_{f'}(n) =$

$\displaystyle \sum_{x\leq n \leq 2x: n = b\ (W)} \sigma_{\tilde f,k-1}(n) \sigma_{\tilde f',k-1}(n) \tilde \theta_{f_k,\tilde f_k}(n+h_k) \ \ \ \ \ (8)$

where

$\displaystyle \tilde \theta_{f_k,\tilde f_k}(n) := (\sum_{d|n} \mu(d) f_k(\frac{\log d}{\log R})) (\sum_{d'|n} \mu(d') f'_k(\frac{\log d'}{\log R})).$

One can write ${\tilde \theta_{f_k,\tilde f_k}}$ as a Dirichlet convolution ${\gamma*1}$, where

$\displaystyle \gamma(n) := \sum_{d,d': [d,d'] = n} \mu(d) \mu(d') f_k(\frac{\log d}{\log R}) f'_k( \frac{\log d'}{\log R} ).$

As ${Q+Q'}$ has ${t_k}$ component in ${[0,\frac{2}{\theta}-\epsilon]}$, ${\gamma}$ is supported on values of ${n}$ that are at most ${R^{\frac{2}{\theta}-\epsilon}= x^{1-\epsilon \theta/2}}$. Assuming an Elliott-Halberstam conjecture ${EH[\theta]}$ for such Dirichlet convolutions, we see that ${\tilde \theta_{f_k,\tilde f_k}(n)}$ is well-distributed in most residue classes of modulus up to ${x^\theta = R^2}$, and one can then obtain asymptotics for (8) by a computation very similar to that used to establish (7); we omit the details.