You are currently browsing the monthly archive for December 2013.
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
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
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 are a set of classical estimates concerning the asymptotic distribution of the prime numbers:
Theorem 1 (Mertens’ theorems) In the asymptotic limit , we have
where is the Euler-Mascheroni constant, defined by requiring that
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
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).
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.
(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.
Recent Comments