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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.

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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.

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.

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This is the third 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 330}. Assuming Elliott-Halberstam, {H_2 \leq 330}.
  • (Polymath8b, tentative) {H_2 \leq 484{,}126}. Assuming Elliott-Halberstam, {H_4 \leq 493{,}408}.
  • (Polymath8b) {H_m \leq \exp( 3.817 m )} for sufficiently large {m}. Assuming Elliott-Halberstam, {H_m \ll e^{2m} m \log m} for sufficiently large {m}.

Much of the current focus of the Polymath8b project is on the quantity

\displaystyle M_k = M_k({\cal R}_k) := \sup_F \frac{\sum_{m=1}^k J_k^{(m)}(F)}{I_k(F)}

where {F} ranges over square-integrable functions on the simplex

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

with {I_k, J_k^{(m)}} being the quadratic forms

\displaystyle I_k(F) := \int_{{\cal R}_k} F(t_1,\ldots,t_k)^2\ dt_1 \ldots dt_k

and

\displaystyle J_k^{(m)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_k)\ dt_m)^2

\displaystyle dt_1 \ldots dt_{m-1} dt_{m+1} \ldots dt_k.

It was shown by Maynard that one has {H_m \leq H(k)} whenever {M_k > 4m}, where {H(k)} is the narrowest diameter of an admissible {k}-tuple. As discussed in the previous post, we have slight improvements to this implication, but they are currently difficult to implement, due to the need to perform high-dimensional integration. The quantity {M_k} does seem however to be close to the theoretical limit of what the Selberg sieve method can achieve for implications of this type (at the Bombieri-Vinogradov level of distribution, at least); it seems of interest to explore more general sieves, although we have not yet made much progress in this direction.

The best asymptotic bounds for {M_k} we have are

\displaystyle \log k - \log\log\log k + O(1) \leq M_k \leq \frac{k}{k-1} \log k \ \ \ \ \ (1)

 

which we prove below the fold. The upper bound holds for all {k > 1}; the lower bound is only valid for sufficiently large {k}, and gives the upper bound {H_m \ll e^{2m} \log m} on Elliott-Halberstam.

For small {k}, the upper bound is quite competitive, for instance it provides the upper bound in the best values

\displaystyle 1.845 \leq M_4 \leq 1.848

and

\displaystyle 2.001162 \leq M_5 \leq 2.011797

we have for {M_4} and {M_5}. The situation is a little less clear for medium values of {k}, for instance we have

\displaystyle 3.95608 \leq M_{59} \leq 4.148

and so it is not yet clear whether {M_{59} > 4} (which would imply {H_1 \leq 300}). See this wiki page for some further upper and lower bounds on {M_k}.

The best lower bounds are not obtained through the asymptotic analysis, but rather through quadratic programming (extending the original method of Maynard). This has given significant numerical improvements to our best bounds (in particular lowering the {H_1} bound from {600} to {330}), but we have not yet been able to combine this method with the other potential improvements (enlarging the simplex, using MPZ distributional estimates, and exploiting upper bounds on two-point correlations) due to the computational difficulty involved.

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This is the second 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) {H_1 \leq 600}.
  • (Polymath8b, tentative) {H_2 \leq 484,276}.
  • (Polymath8b, tentative) {H_m \leq \exp( 3.817 m )} for sufficiently large {m}.
  • (Maynard) Assuming the Elliott-Halberstam conjecture, {H_1 \leq 12}, {H_2 \leq 600}, and {H_m \ll m^3 e^{2m}}.

Following the strategy of Maynard, the bounds on {H_m} proceed by combining four ingredients:

  1. Distribution estimates {EH[\theta]} or {MPZ[\varpi,\delta]} for the primes (or related objects);
  2. Bounds for the minimal diameter {H(k)} of an admissible {k}-tuple;
  3. Lower bounds for the optimal value {M_k} to a certain variational problem;
  4. Sieve-theoretic arguments to convert the previous three ingredients into a bound on {H_m}.

Accordingly, the most natural routes to improve the bounds on {H_m} 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) {EH[\theta]} is true for all {0 < \theta < 1/2}.
  • (Polymath8a) {MPZ[\varpi,\delta]} is true for {\frac{600}{7} \varpi + \frac{180}{7}\delta < 1}.
  • (Polymath8a, tentative) {MPZ[\varpi,\delta]} is true for {\frac{1080}{13} \varpi + \frac{330}{13} \delta < 1}.
  • (Elliott-Halberstam conjecture) {EH[\theta]} is true for all {0 < \theta < 1}.

Ingredient 2 was also studied intensively in Polymath8a, and is more or less a solved problem for the values of {k} of interest (with exact values of {H(k)} for {k \leq 342}, and quite good upper bounds for {H(k)} for {k < 5000}, 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

\displaystyle  M_k({\cal R}_k) := \sup_F \frac{\sum_{m=1}^k J_k^{(m)}(F)}{I_k(F)}

for {F: {\cal R}_k \rightarrow {\bf R}} bounded measurable functions, not identically zero, on the simplex

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

with {I_k, J_k^{(m)}} being the quadratic forms

\displaystyle  I_k(F) := \int_{{\cal R}_k} F(t_1,\ldots,t_k)^2\ dt_1 \ldots dt_k

and

\displaystyle  J_k^{(m)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_k)\ dt_i)^2 dt_1 \ldots dt_{m-1} dt_{m+1} \ldots dt_k.

Equivalently, one has

\displaystyle  M_k({\cal R}_k) := \sup_F \frac{\int_{{\cal R}_k} F {\cal L}_k F}{\int_{{\cal R}_k} F^2}

where {{\cal L}_k: L^2({\cal R}_k) \rightarrow L^2({\cal R}_k)} is the positive semi-definite bounded self-adjoint operator

\displaystyle  {\cal L}_k F(t_1,\ldots,t_k) = \sum_{m=1}^k \int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_{m-1},s,t_{m+1},\ldots,t_k)\ ds,

so {M_k} is the operator norm of {{\cal L}}. Another interpretation of {M_k({\cal R}_k)} is that the probability that a rook moving randomly in the unit cube {[0,1]^k} stays in simplex {{\cal R}_k} for {n} moves is asymptotically {(M_k({\cal R}_k)/k + o(1))^n}.

We now have a fairly good asymptotic understanding of {M_k({\cal R}_k)}, with the bounds

\displaystyle  \log k - 2 \log\log k -2 \leq M_k({\cal R}_k) \leq \log k + \log\log k + 2

holding for sufficiently large {k}. There is however still room to tighten the bounds on {M_k({\cal R}_k)} for small {k}; I’ll summarise some of the ideas discussed so far below the fold.

For Ingredient 4, the basic tool is this:

Theorem 1 (Maynard) If {EH[\theta]} is true and {M_k({\cal R}_k) > \frac{2m}{\theta}}, then {H_m \leq H(k)}.

Thus, for instance, it is known that {M_{105} > 4} and {H(105)=600}, and this together with the Bombieri-Vinogradov inequality gives {H_1\leq 600}. 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.

Read the rest of this entry »

For each natural number {m}, let {H_m} denote the quantity

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

where {p_n} denotes the {n\textsuperscript{th}} prime. In other words, {H_m} is the least quantity such that there are infinitely many intervals of length {H_m} that contain {m+1} or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that {H_1 = 2}, and the prime tuples conjecture would imply that {H_m} is equal to the diameter of the narrowest admissible tuple of cardinality {m+1} (thus we conjecturally have {H_1 = 2}, {H_2 = 6}, {H_3 = 8}, {H_4 = 12}, {H_5 = 16}, and so forth; see this web page for further continuation of this sequence).

In 2004, Goldston, Pintz, and Yildirim established the bound {H_1 \leq 16} conditional on the Elliott-Halberstam conjecture, which remains unproven. However, no unconditional finiteness of {H_1} was obtained (although they famously obtained the non-trivial bound {p_{n+1}-p_n = o(\log p_n)}), and even on the Elliot-Halberstam conjecture no finiteness result on the higher {H_m} was obtained either (although they were able to show {p_{n+2}-p_n=o(\log p_n)} on this conjecture). In the recent breakthrough of Zhang, the unconditional bound {H_1 \leq 70,000,000} was obtained, by establishing a weak partial version of the Elliott-Halberstam conjecture; by refining these methods, the Polymath8 project (which I suppose we could retroactively call the Polymath8a project) then lowered this bound to {H_1 \leq 4,680}.

With the very recent preprint of James Maynard, we have the following further substantial improvements:

Theorem 1 (Maynard’s theorem) Unconditionally, we have the following bounds:

  • {H_1 \leq 600}.
  • {H_m \leq C m^3 e^{4m}} for an absolute constant {C} and any {m \geq 1}.

If one assumes the Elliott-Halberstam conjecture, we have the following improved bounds:

  • {H_1 \leq 12}.
  • {H_2 \leq 600}.
  • {H_m \leq C m^3 e^{2m}} for an absolute constant {C} and any {m \geq 1}.

The final conclusion {H_m \leq C m^3 e^{2m}} on Elliott-Halberstam is not explicitly stated in Maynard’s paper, but follows easily from his methods, as I will describe below the fold. (At around the same time as Maynard’s work, I had also begun a similar set of calculations concerning {H_m}, but was only able to obtain the slightly weaker bound {H_m \leq C \exp( C m )} unconditionally.) In the converse direction, the prime tuples conjecture implies that {H_m} should be comparable to {m \log m}. Granville has also obtained the slightly weaker explicit bound {H_m \leq e^{8m+5}} for any {m \geq 1} by a slight modification of Maynard’s argument.

The arguments of Maynard avoid using the difficult partial results on (weakened forms of) the Elliott-Halberstam conjecture that were established by Zhang and then refined by Polymath8; instead, the main input is the classical Bombieri-Vinogradov theorem, combined with a sieve that is closer in spirit to an older sieve of Goldston and Yildirim, than to the sieve used later by Goldston, Pintz, and Yildirim on which almost all subsequent work is based.

The aim of the Polymath8b project is to obtain improved bounds on {H_1, H_2}, and higher values of {H_m}, either conditional on the Elliott-Halberstam conjecture or unconditional. The likeliest routes for doing this are by optimising Maynard’s arguments and/or combining them with some of the results from the Polymath8a project. This post is intended to be the first research thread for that purpose. To start the ball rolling, I am going to give below a presentation of Maynard’s results, with some minor technical differences (most significantly, I am using the Goldston-Pintz-Yildirim variant of the Selberg sieve, rather than the traditional “elementary Selberg sieve” that is used by Maynard (and also in the Polymath8 project), although it seems that the numerology obtained by both sieves is essentially the same). An alternate exposition of Maynard’s work has just been completed also by Andrew Granville.

Read the rest of this entry »

It’s time to (somewhat belatedly) roll over the previous thread on writing the first paper from the Polymath8 project, as this thread is overflowing with comments.  We are getting near the end of writing this large (173 pages!) paper, establishing a bound of 4,680 on the gap between primes, with only a few sections left to thoroughly proofread (and the last section should probably be removed, with appropriate changes elsewhere, in view of the more recent progress by Maynard).  As before, one can access the working copy of the paper at this subdirectory, as well as the rest of the directory, and the plan is to submit the paper to Algebra and Number theory (and the arXiv) once there is consensus to do so.  Even before this paper was submitted, it already has had some impact; Andrew Granville’s exposition of the bounded gaps between primes story for the Bulletin of the AMS follows several of the Polymath8 arguments in deriving the result.

After this paper is done, there is interest in continuing onwards with other Polymath8 – related topics, and perhaps it is time to start planning for them.  First of all, we have an invitation from  the Newsletter of the European Mathematical Society to discuss our experiences and impressions with the project.  I think it would be interesting to collect some impressions or thoughts (both positive and negative)  from people who were highly active in the research and/or writing aspects of the project, as well as from more casual participants who were following the progress more quietly.  This project seemed to attract a bit more attention than most other polymath projects (with the possible exception of the very first project, Polymath1).  I think there are several reasons for this; the project builds upon a recent breakthrough (Zhang’s paper) that attracted an impressive amount of attention and publicity; the objective is quite easy to describe, when compared against other mathematical research objectives; and one could summarise the current state of progress by a single natural number H, which implied by infinite descent that the project was guaranteed to terminate at some point, but also made it possible to set up a “scoreboard” that could be quickly and easily updated.  From the research side, another appealing feature of the project was that – in the early stages of the project, at least – it was quite easy to grab a new world record by means of making a small observation, which made it fit very well with the polymath spirit (in which the emphasis is on lots of small contributions by many people, rather than a few big contributions by a small number of people).  Indeed, when the project first arose spontaneously as a blog post of Scott Morrrison over at the Secret Blogging Seminar, I was initially hesitant to get involved, but soon found the “game” of shaving a few thousands or so off of H to be rather fun and addictive, and with a much greater sense of instant gratification than traditional research projects, which often take months before a satisfactory conclusion is reached.  Anyway, I would welcome other thoughts or impressions on the projects in the comments below (I think that the pace of comments regarding proofreading of the paper has slowed down enough that this post can accommodate both types of comments comfortably.)

Then of course there is the “Polymath 8b” project in which we build upon the recent breakthroughs of James Maynard, which have simplified the route to bounded gaps between primes considerably, bypassing the need for any Elliott-Halberstam type distribution results beyond the Bombieri-Vinogradov theorem.  James has kindly shown me an advance copy of the preprint, which should be available on the arXiv in a matter of days; it looks like he has made a modest improvement to the previously announced results, improving k_0 a bit to 105 (which then improves H to the nice round number of 600).  He also has a companion result on bounding gaps p_{n+m}-p_n between non-consecutive primes for any m (not just m=1), with a bound of the shape H_m := \lim \inf_{n \to \infty} p_{n+m}-p_n \ll m^3 e^{4m}, which is in fact the first time that the finiteness of this limit inferior has been demonstrated.  I plan to discuss these results (from a slightly different perspective than Maynard) in a subsequent blog post kicking off the Polymath8b project, once Maynard’s paper has been uploaded.  It should be possible to shave the value of H = H_1 down further (or to get better bounds for H_m for larger m), both unconditionally and under assumptions such as the Elliott-Halberstam conjecture, either by performing more numerical or theoretical optimisation on the variational problem Maynard is faced with, and also by using the improved distributional estimates provided by our existing paper; again, I plan to discuss these issues in a subsequent post. ( James, by the way, has expressed interest in participating in this project, which should be very helpful.)

Once again it is time to roll over the previous discussion thread, which has become rather full with comments.  The paper is nearly finished (see also the working copy at this subdirectory, as well as the rest of the directory), but several people are carefully proofreading various sections of the paper.  Once all the people doing so have signed off on it, I think we will be ready to submit (there appears to be no objection to the plan to submit to Algebra and Number Theory).

Another thing to discuss is an invitation to Polymath8 to write a feature article (up to 8000 words or 15 pages) for the Newsletter of the European Mathematical Society on our experiences with this project.  It is perhaps premature to actually start writing this article before the main research paper is finalised, but we can at least plan how to write such an article.  One suggestion, proposed by Emmanuel, is to have individual participants each contribute a brief account of their interaction with the project, which we would compile together with some additional text summarising the project as a whole (and maybe some speculation for any lessons we can apply here for future polymath projects).   Certainly I plan to have a separate blog post collecting feedback on this project once the main writing is done.

The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments.  We are still writing the paper, but it appears to have stabilised in a near-final form (source files available here); the main remaining tasks are proofreading, checking the mathematics, and polishing the exposition.  We also have a tentative consensus to submit the paper to Algebra and Number Theory when the proofreading is all complete.

The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not  need any of the advanced exponential sum estimates).  The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.

The main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments.    As with the previous thread, the main focus on the comments to this thread are concerned with writing up the results of the Polymath8 “bounded gaps between primes” project; the latest files on this writeup may be found at this directory, with the most recently compiled PDF file (clocking in at about 90 pages so far, with a few sections still to be written!) being found here.  There is also still some active discussion on improving the numerical results, with a particular focus on improving the sieving step that converts distribution estimates such as MPZ^{(i)}[\varpi,\delta] into weak prime tuples results DHL[k_0,2].  (For a discussion of the terminology, and for a general overview of the proof strategy, see this previous progress report on the Polymath8 project.)  This post can also contain any other discussion pertinent to any aspect of the polymath8 project, of course.

There are a few sections that still need to be written for the draft, mostly concerned with the Type I, Type II, and Type III estimates.  However, the proofs of these estimates exist already on this blog, so I hope to transcribe them to the paper fairly shortly (say by the end of this week).  Barring any unexpected surprises, or major reorganisation of the paper, it seems that the main remaining task in the writing process would be the proofreading and polishing, and turning from the technical mathematical details to expository issues.  As always, feedback from casual participants, as well as those who have been closely involved with the project, would be very valuable in this regard.  (One small comment, by the way, regarding corrections: as the draft keeps changing with time, referring to a specific line of the paper using page numbers and line numbers can become inaccurate, so if one could try to use section numbers, theorem numbers, or equation numbers as reference instead (e.g. “the third line after (5.35)” instead of “the twelfth line of page 54″) that would make it easier to track down specific portions of the paper.)

Also, we have set up a wiki page for listing the participants of the polymath8 project, their contact information, and grant information (if applicable).  We have two lists of participants; one for those who have been making significant contributions to the project (comparable to that of a co-author of a traditional mathematical research paper), and another list for those who have made auxiliary contributions (e.g. typos, stylistic suggestions, or supplying references) that would typically merit inclusion in the Acknowledgments section of a traditional paper.  It’s difficult to exactly draw the line between the two types of contributions, but we have relied in the past on self-reporting, which has worked pretty well so far.  (By the time this project concludes, I may go through the comments to previous posts and see if any further names should be added to these lists that have not already been self-reported.)

 

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