Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further

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 ;

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 ;

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 ).

Polymath8b, VI: A low dimensional variational problem

This is the sixth 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 (which has recently returned to full functionality, after a partial outage).

The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to , which looks to be the limit of the method (see this previous comment for a semi-rigorous reason as to why 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 with quantities , and a non-trivial square-integrable function supported on such that

• 
• when ;
• when ;
• when ;

and such that we have the inequality

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

An affirmative answer to this question will imply on GEH. We are “within almost 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.

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

with symmetric as well, and (we suspect that the optimal will be roughly ). (However, it is certainly worth also taking a look at easier model problems, such as the polytope , which has no vanishing marginal conditions to contend with; more recently we have been looking at the non-convex polytope .) 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 (currently at ), and on any further progress in asymptotic bounds for for larger – but the current focus is certainly on the bound on on GEH, as we seem to be tantalisingly close to an optimal result here.

Polymath8b, V: Stretching the sieve support further

This is the fifth 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 (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for has been shaved a little from to , and we have very recently achieved the bound 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 for , both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for as .

The basic strategy for bounding still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible -tuple , one needs to locate a non-negative sieve weight , supported on an interval for a large , such that the ratio

is asymptotically larger than as ; this will show that . Thus one wants to locate a sieve weight 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 to the numerator, or by subtracting the term from the numerator (which allows one to reduce the bound to ); 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

where is the divisor sum

with , is the level of distribution ( if relying on Bombieri-Vinogradov, if assuming Elliott-Halberstam, and (in principle) if using Polymath8a technology), and is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions 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 is supported on the simplex

and we define the mixed partial derivative by

then the denominator in (1) is

where

and

Similarly, the numerator of (1) is

where

Thus, if we let be the supremum of the ratio

whenever is supported on and is non-vanishing, then one can prove whenever

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 to the generalised Elliott-Halberstam hypothesis (currently known for , thanks to Motohashi, but conjectured for ), the asymptotic (2) holds under the more general hypothesis that is supported in a polytope , as long as obeys the inclusion

examples of polytopes obeying this constraint include the modified simplex

the prism

the dilated simplex

and the truncated simplex

See this previous post for a proof of these claims.

With regards to the numerator, the asymptotic (3) is valid whenever, for each , the marginals vanish outside of . This is automatic if is supported on , or on the slightly larger region , but is an additional constraint when is supported on one of the other polytopes mentioned above.

More recently, we have obtained a more flexible version of the above asymptotic: if the marginals vanish outside of for some , then the numerator of (1) has a lower bound of

where

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

Theorem 1 Suppose we can find and a function supported on a polytope obeying (4), not identically zero and with all marginals vanishing outside of , and with

Then implies .

In principle, this very flexible criterion for upper bounding should lead to better bounds than before, and in particular we have now established on GEH.

Another promising direction is to try to improve the analysis at medium (more specifically, in the regime ), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using and using the baseline analysis, basically for two reasons:

• We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex in medium dimension.
• The estimates produced by Polymath8a involve a parameter, which introduces additional restrictions on the support of (conservatively, it restricts to where and ; 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 should be possible.

A sporadic image resizing bug in WordPress LaTeX

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.)

Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics

This is the fourth 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) Assuming the Elliott-Halberstam conjecture, .
• (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
• (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
• (Polymath8b, tentative) . (Presumably a comparable bound also holds for on Elliott-Halberstam, but this has not been computed.)
• (Polymath8b) for sufficiently large . Assuming Elliott-Halberstam, for sufficiently large .

While the 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 . This is because we have begun to exploit more fully the fact that when using “multidimensional Selberg-GPY” sieves of the form

with

where , it is not necessary for the smooth function to be supported on the simplex

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

However, it turns out that more is true: given a sufficiently general version of the Elliott-Halberstam conjecture at the given value of , one may work with functions supported on more general domains , so long as the sumset is contained in the non-convex region

and also provided that the restriction

is supported on the simplex

More precisely, if is a smooth function, not identically zero, with the above properties for some , and the ratio

is larger than , then the claim holds (assuming ), and in particular .

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 , the upshot being that if one can find a smooth function supported on that obeys the vanishing marginal conditions

whenever and , and the ratio

is larger than , where

and

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

We are now exploring various choices of to work with, including the prism

and the symmetric region

By suitably subdividing these regions into polytopes, and working with piecewise polynomial functions 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 . Extending this program to , there is a decent chance that we will be able to obtain on EH.

We have also been able to numerically optimise quite accurately for medium values of (e.g. ), which has led to improved values of without EH. For large , we now also have the asymptotic with explicit error terms (details here) which have allowed us to slightly improve the numerology, and also to get explicit numerology for the first time.

Mertens’ theorems

Mertens’ theorems are a set of classical estimates concerning the asymptotic distribution of the prime numbers:

Theorem 1 (Mertens’ theorems) In the asymptotic limit , we have

and

where is the Euler-Mascheroni constant, defined by requiring that

in the limit .

The third theorem (3) is usually stated in exponentiated form

but in the logarithmic form (3) we see that it is strictly stronger than (2), in view of the asymptotic .

Remarkably, these theorems can be proven without the assistance of the prime number theorem

which was proven about two decades after Mertens’ work. (But one can certainly use versions of the prime number theorem with good error term, together with summation by parts, to obtain good estimates on the various errors in Mertens’ theorems.) Roughly speaking, the reason for this is that Mertens’ theorems only require control on the Riemann zeta function in the neighbourhood of the pole at , whereas (as discussed in this previous post) the prime number theorem requires control on the zeta function on (a neighbourhood of) the line . Specifically, Mertens’ theorem is ultimately deduced from the Euler product formula

valid in the region (which is ultimately a Fourier-Dirichlet transform of the fundamental theorem of arithmetic), and following crude asymptotics:

Proposition 2 (Simple pole) For sufficiently close to with , we have

and

Proof: For as in the proposition, we have for any natural number and , and hence

Summing in and using the identity , we obtain the first claim. Similarly, we have

and by summing in and using the identity (the derivative of the previous identity) we obtain the claim.

The first two of Mertens’ theorems (1), (2) are relatively easy to prove, and imply the third theorem (3) except with replaced by an unspecified absolute constant. To get the specific constant requires a little bit of additional effort. From (4), one might expect that the appearance of arises from the refinement

that one can obtain to (6). However, it turns out that the connection is not so much with the zeta function, but with the Gamma function, and specifically with the identity (which is of course related to (7) through the functional equation for zeta, but can be proven without any reference to zeta functions). More specifically, we have the following asymptotic for the exponential integral:

Proposition 3 (Exponential integral asymptotics) For sufficiently small , one has

A routine integration by parts shows that this asymptotic is equivalent to the identity

which is the identity mentioned previously.

Proof: We start by using the identity to express the harmonic series as

or on summing the geometric series

Since , we thus have

making the change of variables , this becomes

As , converges pointwise to and is pointwise dominated by . Taking limits as using dominated convergence, we conclude that

or equivalently

The claim then follows by bounding the portion of the integral on the left-hand side.

Below the fold I would like to record how Proposition 2 and Proposition 3 imply Theorem 1; the computations are utterly standard, and can be found in most analytic number theory texts, but I wanted to write them down for my own benefit (I always keep forgetting, in particular, how the third of Mertens’ theorems is proven).

Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves

This is the third 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) Assuming the Elliott-Halberstam conjecture, .
• (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
• (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
• (Polymath8b) for sufficiently large . Assuming Elliott-Halberstam, for sufficiently large .

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

where ranges over square-integrable functions on the simplex

with being the quadratic forms

and

It was shown by Maynard that one has whenever , where is the narrowest diameter of an admissible -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 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 we have are

 

which we prove below the fold. The upper bound holds for all ; the lower bound is only valid for sufficiently large , and gives the upper bound on Elliott-Halberstam.

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

and

we have for and . The situation is a little less clear for medium values of , for instance we have

and so it is not yet clear whether (which would imply ). See this wiki page for some further upper and lower bounds on .

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 bound from to ), 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.

Ultraproducts as a Bridge Between Discrete and Continuous Analysis

(This is an extended blog post version of my talk “Ultraproducts as a Bridge Between Discrete and Continuous Analysis” that I gave at the Simons institute for the theory of computing at the workshop “Neo-Classical methods in discrete analysis“. Some of the material here is drawn from previous blog posts, notably “Ultraproducts as a bridge between hard analysis and soft analysis” and “Ultralimit analysis and quantitative algebraic geometry“‘. The text here has substantially more details than the talk; one may wish to skip all of the proofs given here to obtain a closer approximation to the original talk.)

Discrete analysis, of course, is primarily interested in the study of discrete (or “finitary”) mathematical objects: integers, rational numbers (which can be viewed as ratios of integers), finite sets, finite graphs, finite or discrete metric spaces, and so forth. However, many powerful tools in mathematics (e.g. ergodic theory, measure theory, topological group theory, algebraic geometry, spectral theory, etc.) work best when applied to continuous (or “infinitary”) mathematical objects: real or complex numbers, manifolds, algebraic varieties, continuous topological or metric spaces, etc. In order to apply results and ideas from continuous mathematics to discrete settings, there are basically two approaches. One is to directly discretise the arguments used in continuous mathematics, which often requires one to keep careful track of all the bounds on various quantities of interest, particularly with regard to various error terms arising from discretisation which would otherwise have been negligible in the continuous setting. The other is to construct continuous objects as limits of sequences of discrete objects of interest, so that results from continuous mathematics may be applied (often as a “black box”) to the continuous limit, which then can be used to deduce consequences for the original discrete objects which are quantitative (though often ineffectively so). The latter approach is the focus of this current talk.

The following table gives some examples of a discrete theory and its continuous counterpart, together with a limiting procedure that might be used to pass from the former to the latter:

(Discrete)	(Continuous)	(Limit method)
Ramsey theory	Topological dynamics	Compactness
Density Ramsey theory	Ergodic theory	Furstenberg correspondence principle
Graph/hypergraph regularity	Measure theory	Graph limits
Polynomial regularity	Linear algebra	Ultralimits
Structural decompositions	Hilbert space geometry	Ultralimits
Fourier analysis	Spectral theory	Direct and inverse limits
Quantitative algebraic geometry	Algebraic geometry	Schemes
Discrete metric spaces	Continuous metric spaces	Gromov-Hausdorff limits
Approximate group theory	Topological group theory	Model theory

As the above table illustrates, there are a variety of different ways to form a limiting continuous object. Roughly speaking, one can divide limits into three categories:

• Topological and metric limits. These notions of limits are commonly used by analysts. Here, one starts with a sequence (or perhaps a net) of objects in a common space , which one then endows with the structure of a topological space or a metric space, by defining a notion of distance between two points of the space, or a notion of open neighbourhoods or open sets in the space. Provided that the sequence or net is convergent, this produces a limit object , which remains in the same space, and is “close” to many of the original objects with respect to the given metric or topology.
• Categorical limits. These notions of limits are commonly used by algebraists. Here, one starts with a sequence (or more generally, a diagram) of objects in a category , which are connected to each other by various morphisms. If the ambient category is well-behaved, one can then form the direct limit or the inverse limit of these objects, which is another object in the same category , and is connected to the original objects by various morphisms.
• Logical limits. These notions of limits are commonly used by model theorists. Here, one starts with a sequence of objects or of spaces , each of which is (a component of) a model for given (first-order) mathematical language (e.g. if one is working in the language of groups, might be groups and might be elements of these groups). By using devices such as the ultraproduct construction, or the compactness theorem in logic, one can then create a new object or a new space , which is still a model of the same language (e.g. if the spaces were all groups, then the limiting space will also be a group), and is “close” to the original objects or spaces in the sense that any assertion (in the given language) that is true for the limiting object or space, will also be true for many of the original objects or spaces, and conversely. (For instance, if is an abelian group, then the will also be abelian groups for many .)

The purpose of this talk is to highlight the third type of limit, and specifically the ultraproduct construction, as being a “universal” limiting procedure that can be used to replace most of the limits previously mentioned. Unlike the topological or metric limits, one does not need the original objects to all lie in a common space in order to form an ultralimit ; they are permitted to lie in different spaces ; this is more natural in many discrete contexts, e.g. when considering graphs on vertices in the limit when goes to infinity. Also, no convergence properties on the are required in order for the ultralimit to exist. Similarly, ultraproduct limits differ from categorical limits in that no morphisms between the various spaces involved are required in order to construct the ultraproduct.

With so few requirements on the objects or spaces , the ultraproduct construction is necessarily a very “soft” one. Nevertheless, the construction has two very useful properties which make it particularly useful for the purpose of extracting good continuous limit objects out of a sequence of discrete objects. First of all, there is Łos’s theorem, which roughly speaking asserts that any first-order sentence which is asymptotically obeyed by the , will be exactly obeyed by the limit object ; in particular, one can often take a discrete sequence of “partial counterexamples” to some assertion, and produce a continuous “complete counterexample” that same assertion via an ultraproduct construction; taking the contrapositives, one can often then establish a rigorous equivalence between a quantitative discrete statement and its qualitative continuous counterpart. Secondly, there is the countable saturation property that ultraproducts automatically enjoy, which is a property closely analogous to that of compactness in topological spaces, and can often be used to ensure that the continuous objects produced by ultraproduct methods are “complete” or “compact” in various senses, which is particularly useful in being able to upgrade qualitative (or “pointwise”) bounds to quantitative (or “uniform”) bounds, more or less “for free”, thus reducing significantly the burden of “epsilon management” (although the price one pays for this is that one needs to pay attention to which mathematical objects of study are “standard” and which are “nonstandard”). To achieve this compactness or completeness, one sometimes has to restrict to the “bounded” portion of the ultraproduct, and it is often also convenient to quotient out the “infinitesimal” portion in order to complement these compactness properties with a matching “Hausdorff” property, thus creating familiar examples of continuous spaces, such as locally compact Hausdorff spaces.

Ultraproducts are not the only logical limit in the model theorist’s toolbox, but they are one of the simplest to set up and use, and already suffice for many of the applications of logical limits outside of model theory. In this post, I will set out the basic theory of these ultraproducts, and illustrate how they can be used to pass between discrete and continuous theories in each of the examples listed in the above table.

Apart from the initial “one-time cost” of setting up the ultraproduct machinery, the main loss one incurs when using ultraproduct methods is that it becomes very difficult to extract explicit quantitative bounds from results that are proven by transferring qualitative continuous results to the discrete setting via ultraproducts. However, in many cases (particularly those involving regularity-type lemmas) the bounds are already of tower-exponential type or worse, and there is arguably not much to be lost by abandoning the explicit quantitative bounds altogether.

Polymath8b, II: Optimising the variational problem and the sieve

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:

1. Distribution estimates or for the primes (or related objects);
2. Bounds for the minimal diameter of an admissible -tuple;
3. Lower bounds for the optimal value to a certain variational problem;
4. 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

and

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:

Theorem 1 (Maynard) If is true and , then .

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.

Polymath8b: Bounded intervals with many primes, after Maynard

For each natural number , let denote the quantity

where denotes the prime. In other words, is the least quantity such that there are infinitely many intervals of length that contain or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that , and the prime tuples conjecture would imply that is equal to the diameter of the narrowest admissible tuple of cardinality (thus we conjecturally have , , , , , and so forth; see this web page for further continuation of this sequence).

In 2004, Goldston, Pintz, and Yildirim established the bound conditional on the Elliott-Halberstam conjecture, which remains unproven. However, no unconditional finiteness of was obtained (although they famously obtained the non-trivial bound ), and even on the Elliot-Halberstam conjecture no finiteness result on the higher was obtained either (although they were able to show on this conjecture). In the recent breakthrough of Zhang, the unconditional bound 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 .

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:

• .
• for an absolute constant and any .

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

• .
• .
• for an absolute constant and any .

The final conclusion 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 , but was only able to obtain the slightly weaker bound unconditionally.) In the converse direction, the prime tuples conjecture implies that should be comparable to . Granville has also obtained the slightly weaker explicit bound for any 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 , and higher values of , 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.