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This is the seventh 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 ${H_m}$ can be found at the wiki page.

The current focus is on improving the upper bound on ${H_1}$ under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from ${H_1 \leq 8}$ to ${H_1 \leq 6}$. 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 ${R \subset [0,2]^3}$ with quantities ${0 \leq \varepsilon_1,\varepsilon_2,\varepsilon_3 \leq 1}$, and a non-trivial square-integrable function ${F: {\bf R}^3 \rightarrow {\bf R}}$ supported on ${R}$ such that

• ${R + R \subset \{ (x,y,z) \in [0,4]^3: \min(x+y,y+z,z+x) \leq 2 \},}$
• ${\int_0^\infty F(x,y,z)\ dx = 0}$ when ${y+z \geq 1+\varepsilon_1}$;
• ${\int_0^\infty F(x,y,z)\ dy = 0}$ when ${x+z \geq 1+\varepsilon_2}$;
• ${\int_0^\infty F(x,y,z)\ dz = 0}$ when ${x+y \geq 1+\varepsilon_3}$;

and such that we have the inequality

$\displaystyle \int_{y+z \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y,z)\ dx)^2\ dy dz$

$\displaystyle + \int_{z+x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y,z)\ dy)^2\ dz dx$

$\displaystyle + \int_{x+y \leq 1-\varepsilon_3} (\int_{\bf R} F(x,y,z)\ dz)^2\ dx dy$

$\displaystyle > 2 \int_R F(x,y,z)^2\ dx dy dz?$

An affirmative answer to this question will imply ${H_1 \leq 6}$ on GEH. We are “within two percent” of this claim; we cannot quite reach ${2}$ yet, but have got as far as ${1.962998}$. However, we have not yet fully optimised ${F}$ in the above problem. In particular, the simplex

$\displaystyle R = \{ (x,y,z) \in [0,2]^3: x+y+z \leq 3/2 \}$

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 ${R \subset [0,2]^2}$ with quantities ${0 \leq \varepsilon_1,\varepsilon_2 \leq 1}$, and a non-trivial square-integrable function ${F: {\bf R}^2 \rightarrow {\bf R}}$ supported on ${R}$ such that

• ${R + R \subset \{ (x,y) \in [0,4]^2: \min(x,y) \leq 2 \}}$

$\displaystyle = [0,2] \times [0,4] \cup [0,4] \times [0,2],$

• ${\int_0^\infty F(x,y)\ dx = 0}$ when ${y \geq 1+\varepsilon_1}$;
• ${\int_0^\infty F(x,y)\ dy = 0}$ when ${x \geq 1+\varepsilon_2}$;

and such that we have the inequality

$\displaystyle \int_{y \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y)\ dx)^2\ dy$

$\displaystyle + \int_{x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y)\ dy)^2\ dx$

$\displaystyle > 2 \int_R F(x,y)^2\ dx dy?$

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 ${H_1}$ (or more generally ${H_m}$).

This is the sixth 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 ${H_m}$ can be found at the wiki page (which has recently returned to full functionality, after a partial outage).

The current focus is on improving the upper bound on ${H_1}$ under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from ${H_1 \leq 8}$ to ${H_1 \leq 6}$, which looks to be the limit of the method (see this previous comment for a semi-rigorous reason as to why ${H_1 \leq 4}$ is not possible with this method). With the most general Selberg sieve available, the problem reduces to the following three-dimensional variational one:

Problem 1 Does there exist a (not necessarily convex) polytope ${R \subset [0,1]^3}$ with quantities ${0 \leq \varepsilon_1,\varepsilon_2,\varepsilon_3 \leq 1}$, and a non-trivial square-integrable function ${F: {\bf R}^3 \rightarrow {\bf R}}$ supported on ${R}$ such that

• ${R + R \subset \{ (x,y,z) \in [0,2]^3: \min(x+y,y+z,z+x) \leq 2 \},}$
• ${\int_0^\infty F(x,y,z)\ dx = 0}$ when ${y+z \geq 1+\varepsilon_1}$;
• ${\int_0^\infty F(x,y,z)\ dy = 0}$ when ${x+z \geq 1+\varepsilon_2}$;
• ${\int_0^\infty F(x,y,z)\ dz = 0}$ when ${x+y \geq 1+\varepsilon_3}$;

and such that we have the inequality

$\displaystyle \int_{y+z \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y,z)\ dx)^2\ dy dz$

$\displaystyle + \int_{z+x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y,z)\ dy)^2\ dz dx$

$\displaystyle + \int_{x+y \leq 1-\varepsilon_3} (\int_{\bf R} F(x,y,z)\ dz)^2\ dx dy$

$\displaystyle > 2 \int_R F(x,y,z)^2\ dx dy dz?$

(Initially it was assumed that ${R}$ was convex, but we have now realised that this is not necessary.)

An affirmative answer to this question will imply ${H_1 \leq 6}$ on GEH. We are “within almost two percent” of this claim; we cannot quite reach ${2}$ yet, but have got as far as ${1.959633}$. However, we have not yet fully optimised ${F}$ in the above problem.

The most promising route so far is to take the symmetric polytope

$\displaystyle R = \{ (x,y,z) \in [0,1]^3: x+y+z \leq 3/2 \}$

with ${F}$ symmetric as well, and ${\varepsilon_1=\varepsilon_2=\varepsilon_3=\varepsilon}$ (we suspect that the optimal ${\varepsilon}$ will be roughly ${1/6}$). (However, it is certainly worth also taking a look at easier model problems, such as the polytope ${{\cal R}'_3 := \{ (x,y,z) \in [0,1]^3: x+y,y+z,z+x \leq 1\}}$, which has no vanishing marginal conditions to contend with; more recently we have been looking at the non-convex polytope ${R = \{x+y,x+z \leq 1 \} \cup \{ x+y,y+z \leq 1 \} \cup \{ x+z,y+z \leq 1\}}$.) Some further details of this particular case are given below the fold.

There should still be some progress to be made in the other regimes of interest – the unconditional bound on ${H_1}$ (currently at ${270}$), and on any further progress in asymptotic bounds for ${H_m}$ for larger ${m}$ – but the current focus is certainly on the bound on ${H_1}$ on GEH, as we seem to be tantalisingly close to an optimal result here.

This is the fifth 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 ${H_m}$ can be found at the wiki page (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for ${H_1}$ has been shaved a little from ${272}$ to ${270}$, and we have very recently achieved the bound ${H_1 \leq 8}$ 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 ${H_m}$ for ${m \leq 5}$, both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for ${H_m}$ as ${m \rightarrow \infty}$.

The basic strategy for bounding ${H_m}$ still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible ${k}$-tuple ${(h_1,\dots,h_k)}$, one needs to locate a non-negative sieve weight ${\nu: {\bf Z} \rightarrow {\bf R}^+}$, supported on an interval ${[x,2x]}$ for a large ${x}$, such that the ratio

$\displaystyle \frac{\sum_{i=1}^k \sum_n \nu(n) 1_{n+h_i \hbox{ prime}}}{\sum_n \nu(n)} \ \ \ \ \ (1)$

is asymptotically larger than ${m}$ as ${x \rightarrow \infty}$; this will show that ${H_m \leq h_k-h_1}$. Thus one wants to locate a sieve weight ${\nu}$ 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 ${\sum_n \nu(n) 1_{n+h_1,\dots,n+h_k \hbox{ composite}}}$ to the numerator, or by subtracting the term ${\sum_n \nu(n) 1_{n+h_1,n+h_k \hbox{ prime}}}$ from the numerator (which allows one to reduce the bound ${h_k-h_1}$ to ${\max(h_k-h_2,h_{k-1}-h_1)}$); 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

$\displaystyle \nu(n) := 1_{n=b\ (W)} 1_{[x,2x]}(n) \lambda(n)^2$

where ${\lambda(n)}$ is the divisor sum

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

with ${R = x^{\theta/2}}$, ${\theta}$ is the level of distribution (${\theta=1/2-}$ if relying on Bombieri-Vinogradov, ${\theta=1-}$ if assuming Elliott-Halberstam, and (in principle) ${\theta = \frac{1}{2} + \frac{13}{540}-}$ if using Polymath8a technology), and ${f: [0,+\infty)^k \rightarrow {\bf R}}$ is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions ${f}$ 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 ${f}$ is supported on the simplex

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

and we define the mixed partial derivative ${F: [0,+\infty)^k \rightarrow {\bf R}}$ by

$\displaystyle F(t_1,\dots,t_k) = \frac{\partial^k}{\partial t_1 \dots \partial t_k} f(t_1,\dots,t_k)$

then the denominator in (1) is

$\displaystyle \frac{Bx}{W} (I_k(F) + o(1)) \ \ \ \ \ (2)$

where

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

and

$\displaystyle I_k(F) := \int_{[0,+\infty)^k} F(t_1,\dots,t_k)^2\ dt_1 \dots dt_k.$

Similarly, the numerator of (1) is

$\displaystyle \frac{Bx}{W} \frac{2}{\theta} (\sum_{j=1}^m J^{(m)}_k(F) + o(1)) \ \ \ \ \ (3)$

where

$\displaystyle J_k^{(m)}(F) := \int_{[0,+\infty)^{k-1}} (\int_0^\infty F(t_1,\ldots,t_k)\ dt_m)^2\ dt_1 \dots dt_{m-1} dt_{m+1} \dots dt_k.$

Thus, if we let ${M_k}$ be the supremum of the ratio

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

whenever ${F}$ is supported on ${{\cal R}_k}$ and is non-vanishing, then one can prove ${H_m \leq h_k - h_1}$ whenever

$\displaystyle M_k > \frac{2m}{\theta}.$

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 ${EH[\theta]}$ to the generalised Elliott-Halberstam hypothesis ${GEH[\theta]}$ (currently known for ${\theta < 1/2}$, thanks to Motohashi, but conjectured for ${\theta < 1}$), the asymptotic (2) holds under the more general hypothesis that ${F}$ is supported in a polytope ${R}$, as long as ${R}$ obeys the inclusion

$\displaystyle R + R \subset \bigcup_{m=1}^k \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: \ \ \ \ \ (4)$

$\displaystyle t_1+\dots+t_{m-1}+t_{m+1}+\dots+t_k < 2; t_m < 2/\theta \} \cup \frac{2}{\theta} \cdot {\cal R}_k;$

examples of polytopes ${R}$ obeying this constraint include the modified simplex

$\displaystyle {\cal R}'_k := \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: t_1+\dots+t_{m-1}+t_{m+1}+\dots+t_k < 1$

$\displaystyle \hbox{ for all } 1 \leq m \leq k \},$

the prism

$\displaystyle {\cal R}_{k-1} \times [0, 1/\theta)$

the dilated simplex

$\displaystyle \frac{1}{\theta} \cdot {\cal R}_k$

and the truncated simplex

$\displaystyle \frac{k}{k-1} \cdot {\cal R}_k \cap [0,1/\theta)^k.$

See this previous post for a proof of these claims.

With regards to the numerator, the asymptotic (3) is valid whenever, for each ${1 \leq m \leq k}$, the marginals ${\int_0^\infty F(t_1,\ldots,t_k)\ dt_m}$ vanish outside of ${{\cal R}_{k-1}}$. This is automatic if ${F}$ is supported on ${{\cal R}_k}$, or on the slightly larger region ${{\cal R}'_k}$, but is an additional constraint when ${F}$ is supported on one of the other polytopes ${R}$ mentioned above.

More recently, we have obtained a more flexible version of the above asymptotic: if the marginals ${\int_0^\infty F(t_1,\ldots,t_k)\ dt_m}$ vanish outside of ${(1+\varepsilon) \cdot {\cal R}_{k-1}}$ for some ${0 < \varepsilon < 1}$, then the numerator of (1) has a lower bound of

$\displaystyle \frac{Bx}{W} \frac{2}{\theta} (\sum_{j=1}^m J^{(m)}_{k,\varepsilon}(F) + o(1))$

where

$\displaystyle J_{k,\varepsilon}^{(m)}(F) := \int_{(1-\varepsilon) \cdot {\cal R}_{k-1}} (\int_0^\infty F(t_1,\ldots,t_k)\ dt_m)^2\ dt_1 \dots dt_{m-1} dt_{m+1} \dots dt_k.$

A proof is given here. Putting all this together, we can conclude

Theorem 1 Suppose we can find ${0 \leq \varepsilon < 1}$ and a function ${F}$ supported on a polytope ${R}$ obeying (4), not identically zero and with all marginals ${\int_0^\infty F(t_1,\ldots,t_k)\ dt_m}$ vanishing outside of ${(1+\varepsilon) \cdot {\cal R}_{k-1}}$, and with

$\displaystyle \frac{\sum_{m=1}^k J_{k,\varepsilon}^{(m)}(F)}{I_k(F)} > \frac{2m}{\theta}.$

Then ${GEH[\theta]}$ implies ${H_m \leq h_k-h_1}$.

In principle, this very flexible criterion for upper bounding ${H_m}$ should lead to better bounds than before, and in particular we have now established ${H_1 \leq 8}$ on GEH.

Another promising direction is to try to improve the analysis at medium ${k}$ (more specifically, in the regime ${k \sim 50}$), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using ${\theta=1/2}$ and using the baseline ${M_k}$ analysis, basically for two reasons:

• We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex ${{\cal R}_k}$ in medium dimension.
• The estimates ${MPZ^{(i)}[\varpi,\delta]}$ produced by Polymath8a involve a ${\delta}$ parameter, which introduces additional restrictions on the support of ${F}$ (conservatively, it restricts ${F}$ to ${[0,\delta']^k}$ where ${\delta' := \frac{\delta}{1/4+\varpi}}$ and ${\theta = 1/2 + 2 \varpi}$; 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 ${H_1}$ should be possible.

I’m encountering a sporadic bug over the past few months with the way WordPress renders or displays its LaTeX images on this blog (and occasionally on other WordPress blogs).  On most computers, it seems to work fine, but on some computers, the sizes of images are occasionally way off, leading to extremely distorted and fairly unreadable versions of the images appearing in blog posts and comments.  A sample screenshot (with accompanying HTML source), supplied to me by a reader, can be found here (in which an image whose dimensions should be 321 x 59 are instead being displayed as 552 x 20).  Is anyone else encountering this issue?  The problem sometimes can be resolved by refreshing the page, but not always, so it is a bit unclear where the problem is coming from and how one might mitigate it.  (If nothing else, I can add it to the bug collection post, once it can be reliably replicated.)