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Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper “Long gaps between primes“. This is a followup work to our two previous papers (discussed in this previous post), in which we had simultaneously shown that the maximal gap

between primes up to exhibited a lower bound of the shape

for some function that went to infinity as ; this improved upon previous work of Rankin and other authors, who established the same bound but with replaced by a constant. (Again, see the previous post for a more detailed discussion.)

In our previous papers, we did not specify a particular growth rate for . In my paper with Kevin, Ben, and Sergei, there was a good reason for this: our argument relied (amongst other things) on the inverse conjecture on the Gowers norms, as well as the Siegel-Walfisz theorem, and the known proofs of both results both have ineffective constants, rendering our growth function similarly ineffective. Maynard’s approach ostensibly also relies on the Siegel-Walfisz theorem, but (as shown in another recent paper of his) can be made quite effective, even when tracking -tuples of fairly large size (about for some small ). If one carefully makes all the bounds in Maynard’s argument quantitative, one eventually ends up with a growth rate of shape

on the gaps between primes for large ; this is an unpublished calculation of James’.

In this paper we make a further refinement of this calculation to obtain a growth rate

leading to a bound of the form

for large and some small constant . Furthermore, this appears to be the limit of current technology (in particular, falling short of Cramer’s conjecture that is comparable to ); in the spirit of Erdös’ original prize on this problem, I would like to offer 10,000 USD for anyone who can show (in a refereed publication, of course) that the constant here can be replaced by an arbitrarily large constant .

The reason for the growth rate (3) is as follows. After following the sieving process discussed in the previous post, the problem comes down to something like the following: can one sieve out all (or almost all) of the primes in by removing one residue class modulo for all primes in (say) ? Very roughly speaking, if one can solve this problem with , then one can obtain a growth rate on of the shape . (This is an oversimplification, as one actually has to sieve out a random subset of the primes, rather than all the primes in , but never mind this detail for now.)

Using the quantitative “dense clusters of primes” machinery of Maynard, one can find lots of -tuples in which contain at least primes, for as large as or so (so that is about ). By considering -tuples in arithmetic progression, this means that one can find lots of residue classes modulo a given prime in that capture about primes. In principle, this means that union of all these residue classes can cover about primes, allowing one to take as large as , which corresponds to (3). However, there is a catch: the residue classes for different primes may collide with each other, reducing the efficiency of the covering. In our previous papers on the subject, we selected the residue classes randomly, which meant that we had to insert an additional logarithmic safety margin in expected number of times each prime would be shifted out by one of the residue classes, in order to guarantee that we would (with high probability) sift out most of the primes. This additional safety margin is ultimately responsible for the loss in (2).

The main innovation of this paper, beyond detailing James’ unpublished calculations, is to use ideas from the literature on efficient hypergraph covering, to avoid the need for a logarithmic safety margin. The hypergraph covering problem, roughly speaking, is to try to cover a set of vertices using as few “edges” from a given hypergraph as possible. If each edge has vertices, then one certainly needs at least edges to cover all the vertices, and the question is to see if one can come close to attaining this bound given some reasonable uniform distribution hypotheses on the hypergraph . As before, random methods tend to require something like edges before one expects to cover, say of the vertices.

However, it turns out (under reasonable hypotheses on ) to eliminate this logarithmic loss, by using what is now known as the “semi-random method” or the “Rödl nibble”. The idea is to randomly select a small number of edges (a first “nibble”) – small enough that the edges are unlikely to overlap much with each other, thus obtaining maximal efficiency. Then, one pauses to remove all the edges from that intersect edges from this first nibble, so that all remaining edges will not overlap with the existing edges. One then randomly selects another small number of edges (a second “nibble”), and repeats this process until enough nibbles are taken to cover most of the vertices. Remarkably, it turns out that under some reasonable assumptions on the hypergraph , one can maintain control on the uniform distribution of the edges throughout the nibbling process, and obtain an efficient hypergraph covering. This strategy was carried out in detail in an influential paper of Pippenger and Spencer.

In our setup, the vertices are the primes in , and the edges are the intersection of the primes with various residue classes. (Technically, we have to work with a family of hypergraphs indexed by a prime , rather than a single hypergraph, but let me ignore this minor technical detail.) The semi-random method would *in principle* eliminate the logarithmic loss and recover the bound (3). However, there is a catch: the analysis of Pippenger and Spencer relies heavily on the assumption that the hypergraph is uniform, that is to say all edges have the same size. In our context, this requirement would mean that each residue class captures exactly the same number of primes, which is not the case; we only control the number of primes in an average sense, but we were unable to obtain any concentration of measure to come close to verifying this hypothesis. And indeed, the semi-random method, when applied naively, does not work well with edges of variable size – the problem is that edges of large size are much more likely to be eliminated after each nibble than edges of small size, since they have many more vertices that could overlap with the previous nibbles. Since the large edges are clearly the more useful ones for the covering problem than small ones, this bias towards eliminating large edges significantly reduces the efficiency of the semi-random method (and also greatly complicates the *analysis* of that method).

Our solution to this is to iteratively *reweight* the probability distribution on edges after each nibble to compensate for this bias effect, giving larger edges a greater weight than smaller edges. It turns out that there is a natural way to do this reweighting that allows one to repeat the Pippenger-Spencer analysis in the presence of edges of variable size, and this ultimately allows us to recover the full growth rate (3).

To go beyond (3), one either has to find a lot of residue classes that can capture significantly more than primes of size (which is the limit of the multidimensional Selberg sieve of Maynard and myself), or else one has to find a very different method to produce large gaps between primes than the Erdös-Rankin method, which is the method used in all previous work on the subject.

It turns out that the arguments in this paper can be combined with the Maier matrix method to also produce chains of consecutive large prime gaps whose size is of the order of (4); three of us (Kevin, James, and myself) will detail this in a future paper. (A similar combination was also recently observed in connection with our earlier result (1) by Pintz, but there are some additional technical wrinkles required to recover the full gain of (3) for the chains of large gaps problem.)

Kevin Ford, Ben Green, Sergei Konyagin, and myself have just posted to the arXiv our preprint “Large gaps between consecutive prime numbers“. This paper concerns the “opposite” problem to that considered by the recently concluded Polymath8 project, which was concerned with very small values of the prime gap . Here, we wish to consider the *largest* prime gap that one can find in the interval as goes to infinity.

Finding lower bounds on is more or less equivalent to locating long strings of consecutive composite numbers that are not too large compared to the length of the string. A classic (and quite well known) construction here starts with the observation that for any natural number , the consecutive numbers are all composite, because each , is divisible by some prime , while being strictly larger than that prime . From this and Stirling’s formula, it is not difficult to obtain the bound

A more efficient bound comes from the prime number theorem: there are only primes up to , so just from the pigeonhole principle one can locate a string of consecutive composite numbers up to of length at least , thus

where we use or as shorthand for or .

What about upper bounds? The *Cramér random model* predicts that the primes up to are distributed like a random subset of density . Using this model, Cramér arrived at the conjecture

In fact, if one makes the extremely optimistic assumption that the random model perfectly describes the behaviour of the primes, one would arrive at the even more precise prediction

However, it is no longer widely believed that this optimistic version of the conjecture is true, due to some additional irregularities in the primes coming from the basic fact that large primes cannot be divisible by very small primes. Using the Maier matrix method to capture some of this irregularity, Granville was led to the conjecture that

(note that is slightly larger than ). For comparison, the known upper bounds on are quite weak; unconditionally one has by the work of Baker, Harman, and Pintz, and even on the Riemann hypothesis one only gets down to , as shown by Cramér (a slight improvement is also possible if one additionally assumes the pair correlation conjecture; see this article of Heath-Brown and the references therein).

This conjecture remains out of reach of current methods. In 1931, Westzynthius managed to improve the bound (2) slightly to

which Erdös in 1935 improved to

and Rankin in 1938 improved slightly further to

with . Remarkably, this rather strange bound then proved extremely difficult to advance further on; until recently, the only improvements were to the constant , which was raised to in 1963 by Schönhage, to in 1963 by Rankin, to by Maier and Pomerance, and finally to in 1997 by Pintz.

Erdös listed the problem of making arbitrarily large one of his favourite open problems, even offering (“somewhat rashly”, in his words) a cash prize for the solution. Our main result answers this question in the affirmative:

Theorem 1The bound (3) holds for arbitrarily large .

In principle, we thus have a bound of the form

for some that grows to infinity. Unfortunately, due to various sources of ineffectivity in our methods, we cannot provide any explicit rate of growth on at all.

We decided to announce this result the old-fashioned way, as part of a research lecture; more precisely, Ben Green announced the result in his ICM lecture this Tuesday. (The ICM staff have very efficiently put up video of his talks (and most of the other plenary and prize talks) online; Ben’s talk is here, with the announcement beginning at about 0:48. Note a slight typo in his slides, in that the exponent of in the denominator is instead of .) Ben’s lecture slides may be found here.

By coincidence, an independent proof of this theorem has also been obtained very recently by James Maynard.

I discuss our proof method below the fold.

Suppose one is given a -tuple of distinct integers for some , arranged in increasing order. When is it possible to find infinitely many translates of which consists entirely of primes? The case is just Euclid’s theorem on the infinitude of primes, but the case is already open in general, with the case being the notorious twin prime conjecture.

On the other hand, there are some tuples for which one can easily answer the above question in the negative. For instance, the only translate of that consists entirely of primes is , basically because each translate of must contain an even number, and the only even prime is . More generally, if there is a prime such that meets each of the residue classes , then every translate of contains at least one multiple of ; since is the only multiple of that is prime, this shows that there are only finitely many translates of that consist entirely of primes.

To avoid this obstruction, let us call a -tuple *admissible* if it avoids at least one residue class for each prime . It is easy to check for admissibility in practice, since a -tuple is automatically admissible in every prime larger than , so one only needs to check a finite number of primes in order to decide on the admissibility of a given tuple. For instance, or are admissible, but is not (because it covers all the residue classes modulo ). We then have the famous Hardy-Littlewood prime tuples conjecture:

Conjecture 1 (Prime tuples conjecture, qualitative form)If is an admissible -tuple, then there exists infinitely many translates of that consist entirely of primes.

This conjecture is extremely difficult (containing the twin prime conjecture, for instance, as a special case), and in fact there is *no* explicitly known example of an admissible -tuple with for which we can verify this conjecture (although, thanks to the recent work of Zhang, we know that satisfies the conclusion of the prime tuples conjecture for some , even if we can’t yet say what the precise value of is).

Actually, Hardy and Littlewood conjectured a more precise version of Conjecture 1. Given an admissible -tuple , and for each prime , let denote the number of residue classes modulo that meets; thus we have for all by admissibility, and also for all . We then define the *singular series* associated to by the formula

where is the set of primes; by the previous discussion we see that the infinite product in converges to a finite non-zero number.

We will also need some asymptotic notation (in the spirit of “cheap nonstandard analysis“). We will need a parameter that one should think of going to infinity. Some mathematical objects (such as and ) will be independent of and referred to as *fixed*; but unless otherwise specified we allow all mathematical objects under consideration to depend on . If and are two such quantities, we say that if one has for some fixed , and if one has for some function of (and of any fixed parameters present) that goes to zero as (for each choice of fixed parameters).

Conjecture 2 (Prime tuples conjecture, quantitative form)Let be a fixed natural number, and let be a fixed admissible -tuple. Then the number of natural numbers such that consists entirely of primes is .

Thus, for instance, if Conjecture 2 holds, then the number of twin primes less than should equal , where is the twin prime constant

As this conjecture is stronger than Conjecture 1, it is of course open. However there are a number of partial results on this conjecture. For instance, this conjecture is known to be true if one introduces some additional averaging in ; see for instance this previous post. From the methods of sieve theory, one can obtain an *upper bound* of for the number of with all prime, where depends only on . Sieve theory can also give analogues of Conjecture 2 if the primes are replaced by a suitable notion of almost prime (or more precisely, by a weight function concentrated on almost primes).

Another type of partial result towards Conjectures 1, 2 come from the results of Goldston-Pintz-Yildirim, Motohashi-Pintz, and of Zhang. Following the notation of this recent paper of Pintz, for each , let denote the following assertion (DHL stands for “Dickson-Hardy-Littlewood”):

Conjecture 3() Let be a fixed admissible -tuple. Then there are infinitely many translates of which containat least twoprimes.

This conjecture gets harder as gets smaller. Note for instance that would imply all the cases of Conjecture 1, including the twin prime conjecture. More generally, if one knew for some , then one would immediately conclude that there are an infinite number of pairs of consecutive primes of separation at most , where is the minimal diameter amongst all admissible -tuples . Values of for small can be found at this link (with denoted in that page). For large , the best upper bounds on have been found by using admissible -tuples of the form

where denotes the prime and is a parameter to be optimised over (in practice it is an order of magnitude or two smaller than ); see this blog post for details. The upshot is that one can bound for large by a quantity slightly smaller than (and the large sieve inequality shows that this is sharp up to a factor of two, see e.g. this previous post for more discussion).

In a key breakthrough, Goldston, Pintz, and Yildirim were able to establish the following conditional result a few years ago:

Theorem 4 (Goldston-Pintz-Yildirim)Suppose that the Elliott-Halberstam conjecture is true for some . Then is true for some finite . In particular, this establishes an infinite number of pairs of consecutive primes of separation .

The dependence of constants between and given by the Goldston-Pintz-Yildirim argument is basically of the form . (UPDATE: as recently observed by Farkas, Pintz, and Revesz, this relationship can be improved to .)

Unfortunately, the Elliott-Halberstam conjecture (which we will state properly below) is only known for , an important result known as the Bombieri-Vinogradov theorem. If one uses the Bombieri-Vinogradov theorem instead of the Elliott-Halberstam conjecture, Goldston, Pintz, and Yildirim were still able to show the highly non-trivial result that there were infinitely many pairs of consecutive primes with (actually they showed more than this; see e.g. this survey of Soundararajan for details).

Actually, the full strength of the Elliott-Halberstam conjecture is not needed for these results. There is a technical specialisation of the Elliott-Halberstam conjecture which does not presently have a commonly accepted name; I will call it the *Motohashi-Pintz-Zhang conjecture* in this post, where is a parameter. We will define this conjecture more precisely later, but let us remark for now that is a consequence of .

We then have the following two theorems. Firstly, we have the following strengthening of Theorem 4:

Theorem 5 (Motohashi-Pintz-Zhang)Suppose that is true for some . Then is true for some .

A version of this result (with a slightly different formulation of ) appears in this paper of Motohashi and Pintz, and in the paper of Zhang, Theorem 5 is proven for the concrete values and . We will supply a self-contained proof of Theorem 5 below the fold, the constants upon those in Zhang’s paper (in particular, for , we can take as low as , with further improvements on the way). As with Theorem 4, we have an inverse quadratic relationship .

In his paper, Zhang obtained for the first time an unconditional advance on :

This is a deep result, building upon the work of Fouvry-Iwaniec, Friedlander-Iwaniec and Bombieri-Friedlander-Iwaniec which established results of a similar nature to but simpler in some key respects. We will not discuss this result further here, except to say that they rely on the (higher-dimensional case of the) Weil conjectures, which were famously proven by Deligne using methods from l-adic cohomology. Also, it was believed among at least some experts that the methods of Bombieri, Fouvry, Friedlander, and Iwaniec were not quite strong enough to obtain results of the form , making Theorem 6 a particularly impressive achievement.

Combining Theorem 6 with Theorem 5 we obtain for some finite ; Zhang obtains this for but as detailed below, this can be lowered to . This in turn gives infinitely many pairs of consecutive primes of separation at most . Zhang gives a simple argument that bounds by , giving his famous result that there are infinitely many pairs of primes of separation at most ; by being a bit more careful (as discussed in this post) one can lower the upper bound on to , and if one instead uses the newer value for one can instead use the bound . (Many thanks to Scott Morrison for these numerics.) UPDATE: These values are now obsolete; see this web page for the latest bounds.

In this post we would like to give a self-contained proof of both Theorem 4 and Theorem 5, which are both sieve-theoretic results that are mainly elementary in nature. (But, as stated earlier, we will not discuss the deepest new result in Zhang’s paper, namely Theorem 6.) Our presentation will deviate a little bit from the traditional sieve-theoretic approach in a few places. Firstly, there is a portion of the argument that is traditionally handled using contour integration and properties of the Riemann zeta function; we will present a “cheaper” approach (which Ben Green and I used in our papers, e.g. in this one) using Fourier analysis, with the only property used about the zeta function being the elementary fact that blows up like as one approaches from the right. To deal with the contribution of small primes (which is the source of the singular series ), it will be convenient to use the “-trick” (introduced in this paper of mine with Ben), passing to a single residue class mod (where is the product of all the small primes) to end up in a situation in which all small primes have been “turned off” which leads to better pseudorandomness properties (for instance, once one eliminates all multiples of small primes, almost all pairs of remaining numbers will be coprime).

In the third Marker lecture, I would like to discuss the recent progress, particularly by Goldston, Pintz, and Yıldırım, on finding small gaps between consecutive primes. (See also the surveys by Goldston-Pintz-Yıldırım, by Green, and by Soundararajan on the subject; the material here is based to some extent on these prior surveys.)

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