In this post we will record a new truncation of the elementary Selberg sieve discussed in this previous post (and also analysed in the context of bounded prime gaps by Graham-Goldston-Pintz-Yildirim and Motohashi-Pintz) that was recently worked out by Janos Pintz, who has kindly given permission to share this new idea with the Polymath8 project. This new sieve decouples the parameter that was present in our previous analysis of Zhang’s argument into two parameters, a quantity that used to measure smoothness in the modulus, but now measures a weaker notion of “dense divisibility” which is what is really needed in the Elliott-Halberstam type estimates, and a second quantity which still measures smoothness but is allowed to be substantially larger than . Through this decoupling, it appears that the type losses in the sieve theoretic part of the argument can be almost completely eliminated (they basically decay exponential in and have only mild dependence on , whereas the Elliott-Halberstam analyhsis is sensitive only to , allowing one to set far smaller than previously by keeping large). This should lead to noticeable gains in the quantity in our analysis.

To describe this new truncation we need to review some notation. As in all previous posts (in particular, the first post in this series), we have an asymptotic parameter going off to infinity, and all quantities here are implicitly understood to be allowed to depend on (or to range in a set that depends on ) unless they are explicitly declared to be fixed. We use the usual asymptotic notation relative to this parameter . To be able to ignore local factors (such as the singular series ), we also use the “-trick” (as discussed in the first post in this series): we introduce a parameter that grows very slowly with , and set .

For any fixed natural number , define an admissible -tuple to be a fixed tuple of distinct integers which avoids at least one residue class modulo for each prime . Our objective is to obtain the following conjecture for as small a value of the parameter as possible:

Conjecture 1 () Let be a fixed admissible -tuple. Then there exist infinitely many translates of that contain at least two primes.

The twin prime conjecture asserts that holds for as small as , but currently we are only able to establish this result for (see this comment). However, with the new truncated sieve of Pintz described in this post, we expect to be able to lower this threshold somewhat.

In previous posts, we deduced from a technical variant of the Elliot-Halberstam conjecture for certain choices of parameters , . We will use the following formulation of :

Conjecture 2 () Let be a fixed -tuple (not necessarily admissible) for some fixed , and let be a primitive residue class. Then

for any fixed , where , are the square-free integers whose prime factors lie in , and is the quantity

and is the set of congruence classes

and is the polynomial

The conjecture is currently known to hold whenever (see this comment and this confirmation). Actually, we can prove a stronger result than in this regime in a couple ways. Firstly, the congruence classes can be replaced by a more general systetm of congruence classes obeying a certain controlled multiplicity axiom; see this post. Secondly, and more importantly for this post, the requirement that the modulus lies in can be relaxed; see below.

To connect the two conjectures, the previously best known implication was the folowing (see Theorem 2 from this post):

Theorem 3 Let , and be such that we have the inequality

where is the first positive zero of the Bessel function , and are the quantities

and

Then implies .

Actually there have been some slight improvements to the quantities ; see the comments to this previous post. However, the main error remains roughly of the order , which limits one from taking too small.

To improve beyond this, the first basic observation is that the smoothness condition , which implies that all prime divisors of are less than , can be relaxed in the proof of . Indeed, if one inspects the proof of this proposition (described in these three previous posts), one sees that the key property of needed is not so much the smoothness, but a weaker condition which we will call (for lack of a better term) dense divisibility:

Definition 4 Let . A positive integer is said to be -densely divisible if for every , one can find a factor of in the interval . We let denote the set of positive integers that are -densely divisible.

Certainly every integer which is -smooth (i.e. has all prime factors at most is also -densely divisible, as can be seen from the greedy algorithm; but the property of being -densely divisible is strictly weaker than -smoothness, which is a fact we shall exploit shortly.

We now define to be the same statement as , but with the condition replaced by the weaker condition . The arguments in previous posts then also establish whenever .

The main result of this post is then the following implication, essentially due to Pintz:

Theorem 5 Let , , , and be such that

where

and

and

Then implies .

This theorem has rather messy constants, but we can isolate some special cases which are a bit easier to compute with. Setting , we see that vanishes (and the argument below will show that we only need rather than ), and we obtain the following slight improvement of Theorem 3:

Theorem 6 Let , and be such that we have the inequality

where

Then implies .

This is a little better than Theorem 3, because the error has size about , which compares favorably with the error in Theorem 3 which is about . This should already give a “cheap” improvement to our current threshold , though it will fall short of what one would get if one fully optimised over all parameters in the above theorem.

Returning to the full strength of Theorem 5, let us obtain a crude upper bound for that is a little simpler to understand. Extending the summation to infinity and using the Taylor series for the exponential, we have

We can crudely bound

and then optimise in to obtain

Because of the factor in the integrand for and , we expect the ratio to be of the order of , although one will need some theoretical or numerical estimates on Bessel functions to make this heuristic more precise. Setting to be something like , we get a good bound here as long as , which at current values of is a mild condition.

Pintz’s argument uses the elementary Selberg sieve, discussed in this previous post, but with a more efficient estimation of the quantity , in particular avoiding the truncation to moduli between and which was the main source of inefficiency in that previous post. The basic idea is to “linearise” the effect of the truncation of the sieve, so that this contribution can be dealt with by the union bound (basically, bounding the contribution of each large prime one at a time). This mostly avoids the more complicated combinatorial analysis that arose in the analytic Selberg sieve, as seen in this previous post.

— 1. Review of previous material —

In this section we collect some results from previous posts which we will need.

We first record an asymptotic for multiplicative functions. For any natural number , define a -dimensional multiplicative function to be a multiplicative function which obeys the asymptotic

for all . The following result is Lemma 8 from this previous post:

Lemma 7 Let . Let be a fixed positive integer, and let be a multiplicative function of dimension . Then for any fixed compactly supported, Riemann-integrable function , and any for some fixed , one has

Next, we record a criterion for , which is Lemma 7 from this previous post:

Lemma 8 (Criterion for DHL) Let . Suppose that for each fixed admissible -tuple and each congruence class such that is coprime to for all , one can find a non-negative weight function , fixed quantities , a quantity , and a fixed positive power of such that one has the upper bound

the lower bound

for all , and the key inequality

holds. Then holds. Here is defined to equal when is prime and otherwise.

— 2. Pintz’s argument —

We can now prove Theorem 5. Fix to obey the hypotheses of this theorem. Let be a congruence class with coprime to for all (this class exists by the admissibility of ). We apply Lemma 8 with

the elementary Selberg sieve defined by

where

are the multiplicative functions

and

the sieve level is given by the formula

is a fixed smooth function supported on , and is a certain subset of to be chosen shortly. We will assume that non-negative and non-increasing on . In this previous post, we considered this sieve with equal to either all of , or the subset consisting of smooth numbers. For now, we will discuss the estimates as far as we can without having to explicitly specify .

We first consider the asymptotic (5). By arguing exactly as in Section 2 (or Section 3) of this previous post, we can write the left-hand side of (5), up to errors of , as

The summand here is non-negative, so we may crudely replace by all of and apply Lemma 7 to obtain (5) with

Now we turn to the more difficult asymptotic (6). The left-hand side expands as

As observed in Section 2 of this previous post, we have

where

and

Now let be a small fixed constant to be chosen later, and suppose the following claim holds:

Claim 1 (Dense divisibility of moduli) Whenever and are non-zero, then either or else .

Then from the Bombieri-Vinogradov theorem (for the moduli) or the hypothesis (for the larger moduli, noting that ) and standard arguments (cf. Proposition 5 of this post) we have

for any fixed and any multiplicative function of a fixed dimension , where ranges only over those integers of the form with non-zero. From this we easily see (arguing as in Section 2 of this previous post) that the contribution of the error term is , and we are left with establishing the lower bound

up to errors of (henceforth referred to as negligible errors).

As in Section 2 of the previous section, we can write the left-hand side as

where is the -dimensional multiplicative function

So we would like to select small enough that Claim (1) holds, but large enough that we can lower bound (9) by up to negligible errors.

Pintz’s idea is to choose to be the set of all elements of with the property that

Let us first verify Claim 1 with this definition. Suppose that is non-zero, then from (8) and the support of there is a multiple of with (in particular ) and . The latter condition implies that either or that , and . In the latter case we see from (10) (applied to ) that

Thus, if with non-zero, this implies that , and that either or that

The latter conclusion implies that is -densely divisible. Indeed, for any we multiply together all the prime divisors of between and one at a time until just before one reaches or exceeds . This must place one at least as large as . Next, one multiplies to this the prime divisors of less than until one reaches or exceeds ; this is possible thanks to (11), and gives a divisor between and as required.

Now we need to obtain a lower bound for (9). If we write

and

(the minus sign being to compensate for the non-positive nature of ) then we have

and thus

Note that this inequality replaces the quadratic expression with a linear expression in the truncation error , which will be more tractable for computing the effect of that error. We may thus lower bound (9) by the difference of

and

In Section 2 of this post it is shown that

which implies that (12) is equal to

up to negligible errors. Similar considerations show that (13) is equal to

up to negligible errors. To upper bound (14), we need to upper bound

For this we need to catalog the ways in which can fail to be in . In order for this to occur, at least one of the following three statements must hold:

• (i) could be divisible by a prime with .
• (ii) could be divisible by a prime with .
• (iii) lies in , but .

We consider the contributions of (i), (ii), (iii) to (14). We begin with the contribution of (i). This is bounded above by

Applying Lemma 7, one can simplify this modulo negligible errors as

which by another application of Lemma 7 is equal to

where we adopt the notation

Applying Mertens’ theorem and summation by parts, this expression is equal up to negligible errors to

where

Now we turn to the contribution of (ii) to (14). This is bounded above by

By Lemma 7 we have

and so we may bound this contribution up to negligible errors by

which by Lemma 7 again is equal (up to negligible errors) to

By definition, . By Mertens’ theorem, we can thus write the above expression up to negligible errors as

Finally, we turn to the contribution of case (iii) to (14). By Proposition 7 we have

so we may bound this contribution up to negligible errors by

where is as in case (iii).

We introduce the quantities

and

so that case (iii) consists of those in such that

From the support of we may also take . This implies that we may factor

for some primes

and some coprime to , with

The contribution of this case may thus be bounded by

Evaluating the inner sum using Lemma 7, we obtain (up to negligible errors)

where is a truncation of :

Again we have . By Mertens’ theorem we may write this (up to negligible errors) as

Putting all this together, we have obtained the lower bound (6) with

where

and

We now place upper bounds on . In this previous post, the bounds

and

for are proven. Thus we have

These are already quite small for , say, which would correspond to .

For we will use the crude estimate

this may surely be improved, but we will not do so here to simplify the exposition. Then we may bound

The point here is that the first term is exponentially decaying in , which can compensate for the second term if which is currently the case in the regime of interest.

One can do a bit better than this. For any parameter , one has

since the left-hand side vanishes for . This gives the bound

If we insert these bounds into (7), send to zero, and optimise in using Theorem 14 from this previous post, we obtain Theorem 5.

The main estimate was already stated as Theorem 17 in the previous post, but we quickly recall the relevant definitions here. As in other posts, we always take to be a parameter going off to infinity, with the usual asymptotic notation associated to this parameter.

Definition 1 (Coefficient sequences) A coefficient sequence is a finitely supported sequence that obeys the bounds

for all , where is the divisor function.

• (i) If is a coefficient sequence and is a primitive residue class, the (signed) discrepancy of in the sequence is defined to be the quantity

  

• (ii) A coefficient sequence is said to be at scale for some if it is supported on an interval of the form .
• (iii) A coefficient sequence at scale is said to be smooth if it takes the form for some smooth function supported on obeying the derivative bounds

  

  for all fixed (note that the implied constant in the notation may depend on ).

For any , let denote the square-free numbers whose prime factors lie in . The main result of this post is then the following result of Zhang:

Theorem 2 (Type III estimate) Let be fixed quantities, and let be quantities such that

and

and

for some fixed . Let be coefficient sequences at scale respectively with smooth. Then for any we have

In fact we have the stronger “pointwise” estimate

for all with and all , and some fixed .

(This is very slightly stronger than previously claimed, in that the condition has been dropped.)

It turns out that Zhang does not exploit any averaging of the factor, and matters reduce to the following:

Theorem 3 (Type III estimate without ) Let be fixed, and let be quantities such that

and

and

for some fixed . Let be smooth coefficient sequences at scales respectively. Then we have

for all and some fixed .

Let us quickly see how Theorem 3 implies Theorem 2. To show (4), it suffices to establish the bound

for all , where denotes a quantity that is independent of (but can depend on other quantities such as ). The left-hand side can be rewritten as

From Theorem 3 we have

where the quantity does not depend on or . Inserting this asymptotic and using crude bounds on (see Lemma 8 of this previous post) we conclude (4) as required (after modifying slightly).

It remains to establish Theorem 3. This is done by a set of tools similar to that used to control the Type I and Type II sums:

• (i) completion of sums;
• (ii) the Weil conjectures and bounds on Ramanujan sums;
• (iii) factorisation of smooth moduli ;
• (iv) the Cauchy-Schwarz and triangle inequalities (Weyl differencing).

The specifics are slightly different though. For the Type I and Type II sums, it was the classical Weil bound on Kloosterman sums that were the key source of power saving; Ramanujan sums only played a minor role, controlling a secondary error term. For the Type III sums, one needs a significantly deeper consequence of the Weil conjectures, namely the estimate of Bombieri and Birch on a three-dimensional variant of a Kloosterman sum. Furthermore, the Ramanujan sums – which are a rare example of sums that actually exhibit better than square root cancellation, thus going beyond even what the Weil conjectures can offer – make a crucial appearance, when combined with the factorisation of the smooth modulus (this new argument is arguably the most original and interesting contribution of Zhang).

— 1. A three-dimensional exponential sum —

The power savings in Zhang’s Type III argument come from good estimates on the three-dimensional exponential sum

defined for positive integer and (or ). The key estimate is

Theorem 4 (Bombieri-Birch bound) Let be square-free. Then for any we have

where is the greatest common divisor of (and we adopt the convention that ). (Here, the denotes a quantity that goes to zero as , rather than as .)

Note that the square root cancellation heuristic predicts as the size for , thus we can achieve better than square root cancellation if has a common factor with that is not shared with . This improvement over the square root heuristic, which is ultimately due to the presence of a Ramanujan sum inside this three-dimensional exponential sum in certain degenerate cases, is crucial to Zhang’s argument.

Proof: Suppose that factors as , thus are coprime. Then we have

(see Lemma 7 of this previous post). From this and the Chinese remainder theorem we see that factorises as

where . Dilating by , we conclude the multiplicative law

Iterating this law, we see that to prove Theorem 4 it suffices to do so in the case when is prime, or more precisely that

We first consider the case when , so our objective is now to show that

In this case we can write as

Making the change of variables , , this becomes

Performing the sums this becomes

where is the Ramanujan sum

Basic Fourier analysis tells us that equals when and when . The expression (6) then follows from direct computation.

Next, suppose that and . Making the change of variables , becomes

Performing the summation, this becomes

For each , the summation is a Kloosterman sum and is thus by the classical Weil bound (Theorem 8 from previous notes). This gives a net estimate of as desired. Similarly if .

The only remaining case is when . Here one cannot proceed purely through Ramanujan and Weil bounds, and we need to invoke the deep result of Bombieri and Birch, proven in Theorem 1 of the the appendix to this paper of Friedlander and Iwaniec. This bound can be proven by applying Deligne’s proof of the Weil conjectures to a certain -function attached to the surface ; an elementary but somewhat lengthy second proof is also given in the above appendix.

To deal with factors such as , the following simple lemma will be useful.

Lemma 5 For any and any we have

in particular

As in the previous theorem, here denotes a quantity that goes to zero as , rather than as .

Note that it is important that the term is excluded from the first sum, otherwise one acquires an additional term. In particular,

Proof: Estimating

we can bound

— 2. Cauchy-Schwarz —

We now prove Theorem 3. The reader may wish to track the exponents involved in the model regime

where is any fixed power of (e.g. , in which case can be slightly larger than ).

Let be as in Theorem 3, and let be a sufficiently small fixed quantity. It will suffice to show that

where does not depend on . We rewrite the left-hand side as

and then apply completion of sums (Lemma 6 from this previous post) to rewrite this expression as the sum of the main term

plus the error terms

and

where is any fixed quantity and

The first term does not depend on , and the third term is clearly acceptable, so it suffices to show that

It will be convenient to reduce to the case when and are coprime. More precisely, it will suffice to prove the following claim:

Proposition 6 Let be fixed, and let

be such that

and

and

for some fixed , and let be smooth coefficient sequences at scale respectively. Then

for some fixed .

Let us now see why the above proposition implies (8). To prove (8), we may of course assume as the claim is trivial otherwise. We can split

for any function of , so that (8) can be written as

which we expand as

In order to apply Proposition (6) we need to modify the , constraints. By Möbius inversion one has

for any function , and similarly for , so by the triangle inequality we may bound the previous expression by

where

We may discard those values of for which is less than one, as the summation is vacuous in that case. We then apply Proposition (6) with replaced by respectively and set equal to , and replaced by and . One can check that all the hypotheses of Proposition 6 are obeyed, so we may bound (12) by

which by the divisor bound is , which is acceptable (after shrinking slightly).

It remains to prove Proposition 6. Continuing (7), the reader may wish to keep in mind the model case

Note from (9), (10) one has

Expanding out the convolution, our task is to show that

As before, our aim is to obtain a power savings better than over the trivial bound of .

The next step is Weyl differencing. We will need a step size which we will optimise in later. We set

we will make the hypothesis that

and save this condition to be verified later.

By shifting by for and then averaging, we may write the left-hand side of (14) as

By the triangle inequality, it thus suffices to show that

Next, we combine the and summations into a single summation over . We first use a Taylor expansion and (15) to write

for any fixed . If is large enough, then the error term will be acceptable, so it suffices to establish (17) with replaced by for any fixed . We can rewrite

where is such that and

Thus we can estimate the left-hand side of (17) by

where

Here we have bounded by .

We will eliminate the expression via Cauchy-Schwarz. Observe from the smoothness of that

and thus

Note that implies . But from (10) we have , so in fact we have . Thus

From the divisor bound, we see that for each fixed there are choices for , thus

From this, (18), and Cauchy-Schwarz, we see that to prove (17) it will suffice to show that

Comparing with the trivial bound of , our task is now to gain a factor of more than over the trivial bound.

We square out (19) as

If we shift by , then relabel by , and use the fact that , we can reduce this to

Next we perform another completion of sums, this time in the variables, to bound

by

for any fixed , where

(the prime is there to distinguish this quantity from in the introduction) and

Making the change of variables and and comparing with(5), we see that

Applying Theorem 4 (and recalling that ) we reduce to showing that

We now choose to be a factor of , thus

for some coprime to . We compute the sum on the left-hand side:

Lemma 7 We have

Proof: We first consider the contribution of the diagonal case . This term may be estimated by

The term gives , while the contribution of the non-zero are acceptable by Lemma 5.

For the non-diagonal case , we see from Lemma 5 that

since , we obtain a bound of from this case as required.

From this lemma, we see that we are done if we can find obeying

as well as the previously recorded condition (16). We can split the condition (21) into three subconditions:

Substituting the definitions (15), (20) of , we can rewrite all of these conditions as lower and upper bounds on . Indeed, (16) follows from (say)

while the other three conditions rearrange to

and

We can combine (23), (24) into a single condition

Also, from (9), (13) we see that this new condition also implies (22). Thus we are done as soon as we find a factor of such that

where

and

From (11) one has

if is sufficiently small. Also, from (11), (9) one also sees that

and . As is -smooth, we can thus find with the desired properties by the greedy algorithm. (In view of Corollary 12 from this previous post, one could also have ensured that has no tiny factors, although this does not seem to be of much actual use in the Type III analysis.)

Theorem 1 (Szemerédi’s theorem in the primes) Let be a subset of the primes of positive relative density, thus . Then contains arbitrarily long arithmetic progressions.

This result was based in part on an earlier paper of Green that handled the case of progressions of length three. With the primes replaced by the integers, this is of course the famous theorem of Szemerédi.

Szemerédi’s theorem has now been generalised in many different directions. One of these is the multidimensional Szemerédi theorem of Furstenberg and Katznelson, who used ergodic-theoretic techniques to show that any dense subset of necessarily contained infinitely many constellations of any prescribed shape. Our main result is to relativise that theorem to the primes as well:

Theorem 2 (Multidimensional Szemerédi theorem in the primes) Let , and let be a subset of the Cartesian power of the primes of positive relative density, thus

Then for any , contains infinitely many “constellations” of the form with and a positive integer.

In the case when is itself a Cartesian product of one-dimensional sets (in particular, if is all of ), this result already follows from Theorem 1, but there does not seem to be a similarly easy argument to deduce the general case of Theorem 2 from previous results. Simultaneously with this paper, an independent proof of Theorem 2 using a somewhat different method has been established by Cook, Maygar, and Titichetrakun.

The result is reminiscent of an earlier result of mine on finding constellations in the Gaussian primes (or dense subsets thereof). That paper followed closely the arguments of my original paper with Ben Green, namely it first enclosed (a W-tricked version of) the primes or Gaussian primes (in a sieve theoretic-sense) by a slightly larger set (or more precisely, a weight function ) of almost primes or almost Gaussian primes, which one could then verify (using methods closely related to the sieve-theoretic methods in the ongoing Polymath8 project) to obey certain pseudorandomness conditions, known as the linear forms condition and the correlation condition. Very roughly speaking, these conditions assert statements of the following form: if is a randomly selected integer, then the events of simultaneously being an almost prime (or almost Gaussian prime) are approximately independent for most choices of . Once these conditions are satisfied, one can then run a transference argument (initially based on ergodic-theory methods, but nowadays there are simpler transference results based on the Hahn-Banach theorem, due to Gowers and Reingold-Trevisan-Tulsiani-Vadhan) to obtain relative Szemerédi-type theorems from their absolute counterparts.

However, when one tries to adapt these arguments to sets such as , a new difficulty occurs: the natural analogue of the almost primes would be the Cartesian square of the almost primes – pairs whose entries are both almost primes. (Actually, for technical reasons, one does not work directly with a set of almost primes, but would instead work with a weight function such as that is concentrated on a set such as , but let me ignore this distinction for now.) However, this set does not enjoy as many pseudorandomness conditions as one would need for a direct application of the transference strategy to work. More specifically, given any fixed , and random , the four events

do not behave independently (as they would if were replaced for instance by the Gaussian almost primes), because any three of these events imply the fourth. This blocks the transference strategy for constellations which contain some right-angles to them (e.g. constellations of the form ) as such constellations soon turn into rectangles such as the one above after applying Cauchy-Schwarz a few times. (But a few years ago, Cook and Magyar showed that if one restricted attention to constellations which were in general position in the sense that any coordinate hyperplane contained at most one element in the constellation, then this obstruction does not occur and one can establish Theorem 2 in this case through the transference argument.) It’s worth noting that very recently, Conlon, Fox, and Zhao have succeeded in removing of the pseudorandomness conditions (namely the correlation condition) from the transference principle, leaving only the linear forms condition as the remaining pseudorandomness condition to be verified, but unfortunately this does not completely solve the above problem because the linear forms condition also fails for (or for weights concentrated on ) when applied to rectangular patterns.

There are now two ways known to get around this problem and establish Theorem 2 in full generality. The approach of Cook, Magyar, and Titichetrakun proceeds by starting with one of the known proofs of the multidimensional Szemerédi theorem – namely, the proof that proceeds through hypergraph regularity and hypergraph removal – and attach pseudorandom weights directly within the proof itself, rather than trying to add the weights to the result of that proof through a transference argument. (A key technical issue is that weights have to be added to all the levels of the hypergraph – not just the vertices and top-order edges – in order to circumvent the failure of naive pseudorandomness.) As one has to modify the entire proof of the multidimensional Szemerédi theorem, rather than use that theorem as a black box, the Cook-Magyar-Titichetrakun argument is lengthier than ours; on the other hand, it is more general and does not rely on some difficult theorems about primes that are used in our paper.

In our approach, we continue to use the multidimensional Szemerédi theorem (or more precisely, the equivalent theorem of Furstenberg and Katznelson concerning multiple recurrence for commuting shifts) as a black box. The difference is that instead of using a transference principle to connect the relative multidimensional Szemerédi theorem we need to the multiple recurrence theorem, we instead proceed by a version of the Furstenberg correspondence principle, similar to the one that connects the absolute multidimensional Szemerédi theorem to the multiple recurrence theorem. I had discovered this approach many years ago in an unpublished note, but had abandoned it because it required an infinite number of linear forms conditions (in contrast to the transference technique, which only needed a finite number of linear forms conditions and (until the recent work of Conlon-Fox-Zhao) a correlation condition). The reason for this infinite number of conditions is that the correspondence principle has to build a probability measure on an entire -algebra; for this, it is not enough to specify the measure of a single set such as , but one also has to specify the measure of “cylinder sets” such as where could be arbitrarily large. The larger gets, the more linear forms conditions one needs to keep the correspondence under control.

With the sieve weights we were using at the time, standard sieve theory methods could indeed provide a finite number of linear forms conditions, but not an infinite number, so my idea was abandoned. However, with my later work with Green and Ziegler on linear equations in primes (and related work on the Mobius-nilsequences conjecture and the inverse conjecture on the Gowers norm), Tamar and I realised that the primes themselves obey an infinite number of linear forms conditions, so one can basically use the primes (or a proxy for the primes, such as the von Mangoldt function ) as the enveloping sieve weight, rather than a classical sieve. Thus my old idea of using the Furstenberg correspondence principle to transfer Szemerédi-type theorems to the primes could actually be realised. In the one-dimensional case, this simply produces a much more complicated proof of Theorem 1 than the existing one; but it turns out that the argument works as well in higher dimensions and yields Theorem 2 relatively painlessly, except for the fact that it needs the results on linear equations in primes, the known proofs of which are extremely lengthy (and also require some of the transference machinery mentioned earlier). The problem of correlations in rectangles is avoided in the correspondence principle approach because one can compensate for such correlations by performing a suitable weighted limit to compute the measure of cylinder sets, with each requiring a different weighted correction. (This may be related to the Cook-Magyar-Titichetrakun strategy of weighting all of the facets of the hypergraph in order to recover pseudorandomness, although our contexts are rather different.)

The main estimate was already stated as Theorem 16 in the previous post, but we quickly recall the relevant definitions here. As in other posts, we always take to be a parameter going off to infinity, with the usual asymptotic notation associated to this parameter.

Definition 1 (Coefficient sequences) A coefficient sequence is a finitely supported sequence that obeys the bounds

for all , where is the divisor function.

• (i) If is a coefficient sequence and is a primitive residue class, the (signed) discrepancy of in the sequence is defined to be the quantity

  

• (ii) A coefficient sequence is said to be at scale for some if it is supported on an interval of the form .
• (iii) A coefficient sequence at scale is said to obey the Siegel-Walfisz theorem if one has

  

  for any , any fixed , and any primitive residue class .

• (iv) A coefficient sequence at scale is said to be smooth if it takes the form for some smooth function supported on obeying the derivative bounds

  

  for all fixed (note that the implied constant in the notation may depend on .

In Lemma 8 of this previous post we established a collection of “crude estimates” which assert, roughly speaking, that for the purposes of averaged estimates one may ignore the factor in (1) and pretend that was in fact . We shall rely frequently on these “crude estimates” without further citation to that precise lemma.

For any , let denote the square-free numbers whose prime factors lie in .

Definition 2 (Singleton congruence class system) Let . A singleton congruence class system on is a collection of primitive residue classes for each , obeying the Chinese remainder theorem property

whenever are coprime. We say that such a system has controlled multiplicity if the

obeys the estimate

for any fixed and any congruence class with .

The main result of this post is then the following:

Theorem 3 (Type I/II estimate) Let be fixed quantities such that

and

and let be coefficient sequences at scales respectively with

and

with obeying a Siegel-Walfisz theorem. Then for any and any singleton congruence class system with controlled multiplicity we have

The proof of this theorem relies on five basic tools:

• (i) the Bombieri-Vinogradov theorem;
• (ii) completion of sums;
• (iii) and the Weil conjectures;
• (iv) factorisation of smooth moduli ; and
• (v) the Cauchy-Schwarz and triangle inequalities (the dispersion method).

For the purposes of numerics, it is the interplay between (ii), (iii), and (v) that drives the final conditions (7), (8). The Weil conjectures are the primary source of power savings ( for some fixed ) in the argument, but they need to overcome power losses coming from completion of sums, and also each use of Cauchy-Schwarz tends to halve any power savings present in one’s estimates. Naively, one could thus expect to get better estimates by relying more on the Weil conjectures, and less on completion of sums and on Cauchy-Schwarz.

— 1. The Bombieri-Vinogradov theorem —

One of the basic distribution results in this area of analytic number theory is the Bombieri-Vinogradov theorem. As first observed by Motohashi, this theorem in fact controls the distribution of a general class of Dirichlet convolutions in congruence classes. We will use the following formulation of this theorem, essentially Theorem 0 of Bombieri-Friedlander-Iwaniec;

Theorem 4 (Bombieri-Vinogradov theorem) Let be such that

and

for some fixed . Let be coefficient sequences at scale respectively. Suppose also that obeys a Siegel-Walfisz theorem.

• (i) (First Bombieri-Vinogradov inequality) We have

  

  for some sufficiently slowly decaying and any fixed .

• (ii) (Second Bombieri-Vinogradov inequality) we have

  

  for some sufficiently slowly decaying and any fixed .

For sake of completeness we now recall the proof of this theorem, following the presentation in Bombieri-Friedlander-Iwaniec. (This is standard material, and experts may immediately skip to the next section.) We first need the large sieve inequality for Dirichlet characters:

Lemma 5 (Large sieve inequality) For any sequence supported on and any one has

whee the summation is over primitive Dirichlet characters of conductor .

Proof: By enlarging we may assume that .

We use the method. By duality, the desired estimate is equivalent to

which is in turn equivalent to

We bound by a Schwartz function whose Fourier transform is supported on ; this is possible for a fixed when . The left-hand side then expands as

The inner sum is when , and zero otherwise thanks to Fourier analysis. The claim follows.

Now we prove part (i) of Theorem 4. By an overspill argument (cf. Lemma 7 of this previous post) it suffices to show that

for any fixed .

Let . By multiplicative Fourier analysis we may write the left-hand side of (11) as

where the inner sum ranges over non-principal Dirichlet characters of modulus , not necessarily primitive. Any such character can be written as where with , and is a primitive Dirichlet character of conductor . The above sum can then be rewritten as

where the summation ranges over primitive characters modulo . Since , we may bound this by

which on performing the summation is bounded by

and then by dyadic decomposition this is bounded by

Let us first consider the contribution of the small moduli, specifically when for some fixed . From the Siegel-Walfisz theorem (3) we have

for any fixed , and then by crude estimates (see Lemma 8 of this previous post) we see that this contribution is acceptable for (11). Thus we may restrict attention to those moduli with for any fixed . On the other hand, from Lemma 5 we have

for any . From crude estimates we have

Since

the claim follows.

Now we prove (ii). We now set for some fixed . By overspill as before, it suffices to show that

By multiplicative Fourier analysis we have

so by splitting into primitive characters as before we may bound the left-hand side of (12) by

Arguing as before, the case is acceptable, so we assume . From crude estimates we have

and

From Lemma 5 and Cauchy-Schwarz we can thus bound

by

as is comparable to , this simplifies to

which is acceptable from the hypotheses on if is chosen large enough depending on . This concludes the proof of Theorem 4.

We remark that an inspection of the proof reveals that the or factors in the threshold for may be replaced by provided that is sufficiently large depending on . However, this refinement of the Bombieri-Vinogradov inequality does not lead to any improvement in the final numerology for our purposes.

— 2. Completion of sums —

At several stages in the argument we will need to consider sums of the form

for some smooth coefficient sequence , sone congruence classes depending on a further parameter , and further coefficients . The completion of sums technique replaces the term here with an exponential phase, leaving one with consideration of exponential sums such as

for various , where for (or ) is the standard character on . More precisely, we have

Lemma 6 (Completion of sums) Let be a smooth coefficient sequence at some scale . Let be a finite set of indices, and for each let be a complex number and be a congruence class for some . we have

for any fixed .

Specialising to the case and , we conclude in particular that

whenever is periodic of degree .

Proof: We rearrange the left-hand side as

Using the Fourier expansion

and rearranging, the left-hand side then becomes

The term of this is the first term on the right-hand side of (13). The terms coming from integers with can be bounded by the second term in (13), bounding crudely by by crude estimates and also using conjugation symmetry when is negative. So it will suffice to show that

(say) when . Writing as in the definition of a smooth coefficient sequence, and applying Poisson summation, the left-hand side is

where is the Fourier transform of . But from the smoothness (4) of and integration by parts one has the bounds

for any fixed , and from the hypothesis we obtain the claim by taking large enough depending on .

We remark that in the absence of cancellation in the exponential sum , the first error term in (13) could be as large as

which is about times as large as the main term in (13). In practice we will apply this lemma with for some fixed , in which case completion of sums will cost a factor of or so in the bounds. However, it is still often desirable to pay this cost in order to exploit cancellation bounds for exponential sum, in particular those coming from the Weil conjectures as described below.

In our applications, the modulus will split into a product of two factors or three factors . The following simple lemma lets us then split exponential phases of the form :

Lemma 7 Suppose that for some coprime , and let be the intersection of the congruence classes and . Then for any integer ,

Similarly, if for coprime and is the intersection of , , and , then

Proof: We just prove the first identity, as the second is similar. Let be an integer such that , and similarly let be an integer such that . Then is equal to mod and mod , and thus equal to mod . Thus

and the claim follows by factoring the exponential.

— 3. The Weil conjectures —

For the purposes of this post, the Weil conjectures (as proven in full generality by Deligne) can be viewed as a black box device to obtain “square root cancellation” for various exponential sums of the form

where is a finite abelian group (i.e. a finite product of cyclic groups) with some additive character and some rational function , basically by obtaining the analogue of the Riemann Hypothesis for a certain zeta function associated to an algebraic variety related to the function . (This is by no means the full strength of the Weil conjectures; amongst other things, one can also twist such sums by multiplicative characters, and also work with more complicated schemes than classical algebraic varieties, though the exponential sum estimates are more difficult to state succinctly as a consequence.) A basic instance of this is Weil’s classical bound on the Kloosterman sum

defined whenever are integers with positive.

Theorem 8 (Weil bound) For any one has

where is the greatest common divisor of .

Proof: (Sketch only; see e.g. Iwaniec-Kowalski for a full proof.) By the Chinese remainder theorem we may reduce to the case when is a power of a prime, then we may also reduce to the case when are coprime to . It then suffices to show that

whenever is an additive character and is the hyperbola

An important trick is then to generalise from to finite dimensional extensions of , and then consider the Kloosterman sums

associated to these extensions, where is the field trace. For non-principal , it is possible to show the explicit formula

for some complex numbers (depending of course on ); this is part of the general theory of -functions associated to algebraic varieties, but can also be established by elementary means (e.g. by establishing a linear recurrence for the ). For principal , of course, one has

Next, one observes the basic identity

as can be seen by computing the kernel and range of the linear map on (this identity is also related to Hilbert’s Theorem 90). From this we have a combinatorial interpretation of the quantity

namely as the number of points on the curve

One can show (e.g. using Stepanov’s method, cf. this previous post) that the number of this points on this curve is equal to , leading to the identity

Studying the asymptotics as , one is led to the conclusion that (this trick to “magically” delete the error is a canonical example of the tensor power trick), and the bound (16) then follows from the case of (17).

In practice, we shall estimate crudely by the divisor bound , and the factor will also be small in applications, so that we do indeed see the square root savings over the trivial bound . For the Type I/II sums, the classical Weil bound is sufficient; but for the Type III sums that we will cover in a subsequent post, the full force of Deligne’s results are needed.

An important remark is that when , we can apply the change of variables and convert the Kloosterman sum into a Ramanujan sum

which enjoys even better cancellation than square root cancellation; in particular it is not difficult to establish the bound

using the Chinese remainder theorem to reduce to the case when is a power of a prime, and then using the divisor bound.

For the Type I and Type II sums we need a more complicated variant of this bound (Lemma 11 of Zhang’s paper):

Lemma 9 Let be square-free natural numbers, let

and let be integers. Then the double Kloosterman sum

obeys the bound

for some with

for . In particular, from Theorem 8, we have

while in the case we have from (18) that

Proof: As are coprime, we may refactor as

for some , and , with

and

for , which in particular implies that as claimed. Using the Chinese remainder theorem, we may now factor as the product of the sums

and

Bounding the first sum trivially by and shifting the third sum by we obtain the claim.

The treatment of the terms in the above analysis are crude, but in applications is often trivial anyway, so it is not essential to obtain the sharpest estimates here.

We can combine this with the method of completion of sums:

Corollary 10 Let be square-free natural numbers with , let be integers, and let be a smooth coefficient sequence at a scale . Then

Proof: Write . By (14) (and overspill) we may bound the left-hand side by

The first term is by Lemma 9, which is acceptable. Another application of Lemma 9 gives

From the divisor bound one has

for any , so from this and Cauchy-Schwarz the net contribution of the second term is

which is acceptable.

— 4. Factoring a smooth number —

We will need to take advantage of the smooth nature of the variable to factor it into two smaller pieces. We need an elementary lemma:

Lemma 11 Let be a quantity of size , and set

(say). Let be fixed. Then, for all but exceptions, all integers have the property that

in particular,

Proof: Suppose that (19) failed, thus

In particular, we see that has at least prime factors, which implies in particular that

On the other hand, we have the standard bound

and the claim now follows from Markov’s inequality.

Corollary 12 (Good factorisation) Let , and let be quantities such that

Let be fixed. Then for all but exceptions, all have a factorisation

where are coprime with

Furthermore has no prime factors less than , thus

where .

The fact that has no tiny (i.e. less than ) prime factors will imply that any two such will typically be coprime to each other with high probability (at least for any fixed ), which is a key technical fact which we will need to exploit later. (The -trick achieves a qualitatively similar effect, but would only give such a claim with probability or maybe for some small if one really optimised it, which is insufficient for the applications at hand.)

Proof: By Lemma 11 we may restrict attention to those for which

Now is the product of distinct primes of size at most , with . Applying the greedy algorithm, one can then find a factor of with

If one then multiplies by all primes of size less than that divide to create , then sets , the claim follows.

— 5. The dispersion method —

We begin the proof of Theorem 3. The reader may wish to track the exponents involved in the model regime

We can restrict to the range

for some sufficiently slowly decaying , since otherwise we may use the Bombieri-Vinogradov theorem (Theorem 4). Thus we need to show that

Let

be an exponent to be optimised later (in many cases, such as (20), it can be set very close to zero). By Corollary 12, outside of a small number of exceptions, we may factor where with , is coprime to , and

and

Let us first dispose of the set of exceptional values of for which the above factorisation is unavailable. From Corollary 12 we have

On the other hand, we have the crude estimate

which when combined with crude estimates leads to the crude upper bound

Applying Cauchy-Schwarz we conclude that

and so gives a negligible contribution to (21) (increasing as necessary).

For the non-exceptional , we arbitrarily select a factorisation of the above form, and apply a dyadic decomposition. We conclude that it suffices to show that

for any fixed , where obey the size conditions

and

Fix . We replace by , and abbreviate and by and respectively, thus our task is to show that

We now split the discrepancy

as the sum of the subdiscrepancies

and

Of the two, the first discrepancy is significantly more difficult to handle. By the triangle inequality, it will suffice to show that

and

We begin with (26), which is easier. For each fixed , it will suffice to show that

as the claim then follows by dividing by and summing using standard estimates (see Lemma 8 of this previous post). But this claim follows from the Bombieri-Vinogradov theorem (Theorem 4), after restricting to the integers coprime to (which does not affect the property of being a coefficient sequence supported at a certain scale, nor does it affect the Siegel-Walfisz theorem).

Now we establish (25). Here we will not take advantage of the summation, and use crude estimates to reduce to showing that

for each individual with , which we now fix. Actually, we will prove the more general statement

whenever are good singleton congruence class systems. Let us see how (28) implies (27). Observe that if is any integer, then and are also good singleton congruence class systems, where consists of the primes with not divisible by (thus lies in when is coprime to ). By (28) we thus have

If we average over the interval for some sufficiently small fixed , we observe that

and similarly (using the crude divisor bound)

and we conclude from the triangle inequality and the bounds on (if is small enough) that

On the other hand, from crude estimates one has

and the claim (27) then follows from Cauchy-Schwarz (noting from the Chinese remainder theorem that the two constraints are equivalent to the single constraint ).

It remains to prove (28). We will use the dispersion method (or Cauchy-Schwarz), playing the two congruence conditions and against each other. We first get rid of the absolute values in (28) by introducing an additional bounded coefficient. More precisely, to prove (28) it suffices to show that

for any bounded real coefficients . We expand out the Dirichlet convolution

then relabel as to rearrange the left-hand side as

We can write for some smooth non-negative coefficient sequence at scale . From crude bounds one has

so by the Cauchy-Schwarz inequality it suffices to show that

for any fixed . (As a sanity check, note that we are still only asking for a savings over the trivial bound.) Expanding out the square, it suffices to show that

where is subject to the same constraints as (thus and for ), and is some quantity that is independent of the choice of congruence classes , , since by replacing the with or vice versa as necessary one can express (29) as a linear combination of terms of the form in (29) in such a way that all the terms cancel out ().

At this stage we need to deal with a technical problem that may share a common factor; fortunately, this event turns out to be negligible (but only thanks to the controlled multiplicity hypothesis (6). More precisely, we split (30) into the coprime case

and the non-coprime case

We first show (32). The basic point here is that because we have previously restricted to have no prime factor smaller than , we can gain a factor of in (32), which is strong enough to overcome logarithmic losses but not losses coming from the divisor bound. To avoid using the divisor bound we will need the controlled multiplicity hypothesis (6) (and this is the only place in the argument where this hypothesis is actually used).

We turn to the details. The quantities must all lie in . We may split and where are coprime. By the Chinese remainder theorem, the constraints and then imply that

so by the triangle inequality and interchange of summation we can bound the left-hand side by

(with understood to lie in and be at most ) which we can write as

where

and

are the multiplicity functions associated to the congruence class systems and .

By the elementary bound and symmetry we may thus bound the left-hand side of (32) by

plus a symmetric term which is treated similarly and will be ignored.

We rearrange the constraints

as a combination of the constraints

and

For fixed obeying the former set of constraints, the obeying the second set of constraints lie in a single congruence class mod by the Chinese remainder theorem. From the controlled multiplicity hypothesis (6) one thus has

and so the left-hand side of (32) has been bounded by

We first dispose of the error term. From the divisor bound we have . For fixed , the sum then can be bounded by , leading to a total contribution of

We would like to bound this by . This is possible if we have

and

for some fixed . But the former bound is immediate from (23), (10), (22), while from (9), (24), (23) we see that the latter bound will follow if we hav

for some fixed . We file away this necessary condition for now and move on, though we note that these conditions are weaker than (22) except in the “Type II” case when are close to .

Having disposed of the error term, the remaining contribution to the left-hand side of (32) that we need to control is

Summing in using crude bounds, this is bounded by

and then by summing in this is in turn bounded by

The term sums to , which is thanks to (23), (22). The main term is then

Now we finally use the fact that has no small factors less than to bound the summation here by

and since grows faster than any power of we see that this error term is also acceptable for (32). This concludes the proof of (32) (contingent of course on the lower bounds (22), (34) on that we will deal with later).

It remains to verify (31). Observe that must be coprime to and coprime to , with , to have a non-zero contribution to the sum. We then rearrange the left-hand side as

note that these inverses in the various rings , , are well-defined thanks to the coprimality hypotheses.

We may write for some . By the triangle inequality, and relabeling as , it thus suffices to show that for any particular

one has

for some independent of the and .

We remark that at this stage we are only needing to gain a factor of over the trivial bound. However, we will now perform the expensive step of completion of sums (Lemma 6), which replaces the factor by an exponential phase at the cost of requiring now a significantly larger gain over the trivial bound. Applying Lemma 6 and Lemma 7, we can write the left-hand side of (36) as the sum of the main term

an error term

where

is an arbitrary small fixed quantity, and is the phase

(here we use the bounded nature of the ); and another error term that can easily be shown to be for any fixed . The term is independent of the and , so it will suffice to show that

for a sufficiently small fixed , and we have dropped all hypotheses on other than magnitude, and we abbreviate as . As noted after Lemma 13, we are no longer asking for just a savings over the trivial bound; we must instead gain a factor of to overcome the summation in .

Although this is not strictly necessary for our analysis, let is confirm that is actually non-trivial in the sense that

Indeed, from (33) and (9) one has

and hence from (24)

and (40) then follows from (37). Note though in the model case (20) with that is close to (for any between and ).

We now split into two cases, one which works when are not too close to , and one which works when are close to .

Theorem 13 (Type I case) If the inequalities

and

hold for some fixed , then (39) holds for a sufficiently small fixed .

The condition (42) represents a border between this case and the Type II case.

Theorem 14 (Type II case) If the inequality

holds, then (39) holds for a sufficiently small fixed .

Both of these theorems are established by using Cauchy-Schwarz to eliminate the absolute values and factors in (39) until one is left with an expression only involving the phase which can then be estimated using the Weil conjectures with a power saving to counteract the loss of , however the use of Cauchy-Schwarz is slightly different in the two cases.

Assuming these theorems, let us now conclude the proof of Theorem 3. First suppose we are in the “Type I” regime when (42) holds for some fixed . Then by (9) we have

which means that the condition (34) is now weaker than (22) and may be omitted. By (7), (8) we can simultaneously obey (22), (34), (41) by setting sufficiently close to zero, and the claim now follows from Theorem 13. (Note that by adding (7) to (8) we certainly have that .)

Now suppose instead that we are in the “Type II” regime where (42) fails for some small , so that by (9) we have

From this we see that we may replace by in (10) and in all of the above analysis. If we set then the conditions (22), (34) are obeyed. Theorem 14 will then give us what we want provided that

which is satisfied for small enough thanks to (8).

In the next two sections we establish Theorem 13 and Theorem 14.

— 6. The Type I sum —

We now prove Theorem 13. It suffices to show that

for any bounded real coefficients (which are vaguely related to the previous coefficients , but this is not important for us here). We rearrange the left-hand side as

From the divisor bound we have

and we may write for some smooth coefficient sequence at scale , so by Cauchy-Schwarz it suffices to show that

(note now we have to gain more than over the trivial bound, rather than just ). We rearrange this as

so by the triangle inequality it suffices to show that

for some fixed . We discard the summation and reduce to showing that

for any .

To prove (44), we isolate the diagonal case and the non-diagonal case . For the diagonal case, we make the crude bound

The contribution of the diagonal case can now be bounded by

Writing we have , and one can estimate this by

which by the divisor bound is of the form

For this to be acceptable we need a bound of the form

which, by (37) is equivalent to

but this follows from (24), (42) for small enough.

Now we treat the non-diagonal case . The key estimate here is

for all non-diagonal . In the model case (20) with , the two terms on the right-hand side are approximately and , which give the desired power saving compared to the trivial bound of since (and in (20), is small, so just about any power saving suffices). As the model case indicates, the first term in (45) is the dominant one in practice.

Assume for the moment that the estimate (45) holds; then the non-diagonal contribution to (44) is

so to conclude (44) we need to show that

and

Using (37), (9) we can rewrite these criteria as

and

respectively. Applying (24), (23), it suffices to verify that

and

but these follow from (10) and (41) (the latter inequality holds with considerable room to spare).

It remains to show (45) in the non-diagonal case . From (38) we may of course assume that

and the left-hand side of (45) expands as

The first two phases

can be combined as

where and is the residue class

and the inverses are with respect to modulus in the first summand and in the second summand. For future reference we note that

Similarly, the two phases

can combine as

where and is the residue class

although the precise value of will not be important for us. The left-hand side of (45) is thus

and hence Lemma 10 and the coprimality of is bounded by

Note that

which is controlled by the first term on the right-hand side of (45). So it remains to show that

We crudely bound by and by . (This is inefficient, but this term is not dominant in the analysis in any case.) By (46) we may bound by , and the claim follows.

— 7. The Type II sum —

We now prove Theorem 14. As before, it suffices to show that

for any bounded real coefficients . We rearrange the left-hand side slightly differently from the previous section:

From we have

and can be restricted to be less than (say), so by Cauchy-Schwarz it suffices to show that

Expanding and using the triangle inequality as in the previous section, we reduce to showing that

This is basically the same situation as in the previous section except that we have decoupled the variables from each other.

As before, we isolate a diagonal case , and now consider the contribution of this case. Arguing as in the previous section, the contribution of this case to (47) can be bounded by

which by the divisor bound is of the form

which will be acceptable if

By (37) this is equivalent to

but this is automatic from (23) and (22).

For the off-diagonal case, we use the following variant

of (45), valid for all non-diagonal . The bound here is weaker than in (45), but in the model case (20) the right-hand side terms are approximately and respectively, which still represents a power saving over the trivial bound of approximately as long as . While this does not cover all the range that the Type I analysis does, it crucially is able to cover the case when is very close to zero, which the Type I analysis does not cover due to the condition (42). The Type I/II border is not the critical border for optimising the exponents, so it is not a priority for us to improve bounds in the Type II analysis such as (48).

We assume (48) for now and finish the proof of Theorem 14 by numerical computations similar to that in the previous section. The non-diagonal contribution to (47) is now estimated by

so to conclude (44) we need to show that

and

Using (37), (9) we can rewrite these criteria as

and

respectively. Applying (24), (23), it suffices to verify that

and

but these follow from (10) and (43).

It remains to prove (48). This is very similar to the treatment of (45). From (38) we may assume

and the left-hand side of (45) expands as

If we set

then as before we can rewrite this sum as

where

(with the inverses with respect to the moduli in the first, second, and third summands respectively), and the value of is unimportant for us. We have an analogue of (46), namely

We apply Lemma 10 as before, although things are not quite as favorable because need not be coprime in this case. This bounds the left-hand side of (48) by

We have

which is controlled by the first term on the right-hand side of (48). So it remains to show that

We crudely bound by and by ; also

and from (49) one has . The claim follows.

As usual, we let be a large asymptotic parameter, and a sufficiently slowly growing function of . Let and be such that holds (see this previous post for a definition of this assertion). We let be a fixed admissible -tuple, let , let be the square-free numbers with prime divisors in , and consider the truncated GPY sieve

where

where , is the polynomial

and is a fixed smooth function supported on . As discussed in the previous post, we are interested in obtaining an upper bound of the form

as well as a lower bound of the form

for all (where when is prime and otherwise), since this will give the conjecture (i.e. infinitely many prime gaps of size at most ) whenever

It turns out we in fact have precise asymptotics

and

although the exact formulae for are a little complicated. (The fact that could be computed exactly was already anticipated in Zhang’s paper; see the remark on page 24.) We proceed as in the previous post. Indeed, from the arguments in that post, (2) is equivalent to

and (3) is similarly equivalent to

Here is the number of prime factors of .

We will work for now with (4), as the treatment of (5) is almost identical.

We would now like to replace the truncated interval with the untruncated interval , where . Unfortunately this replacement was not quite done correctly in the previous post, and this will now be corrected here. We first observe that if is any finitely supported function, then by Möbius inversion we have

Note that if and only if we have a factorisation , with and coprime to , and that this factorisation is unique. From this, we see that we may rearrange the previous expression as

Applying this to (4), and relabeling as , we conclude that the left-hand side of (4) is equal to

which may be rearranged as

This is almost the same formula that we had in the previous post, except that the Möbius function of the greatest common divisor of was missing, and also the coprimality condition was not handled properly in the previous post.

We may now eliminate the condition as follows. Suppose that there is a prime that divides both and . The expression

can then be bounded by

which may be factorised as

which by Mertens’ theorem (or the simple pole of at ) is

Summing over all gives a negligible contribution to (6) for the purposes of (4). Thus we may effectively replace (6) by

The inner summation can be treated using Proposition 10 of the previous post. We can then reduce (4) to

where is the function

Note that vanishes if or . In practice, we will work with functions in which has a definite sign (in our normalisations, will be non-positive), making non-negative.

We rewrite the left-hand side of (7) as

We may factor for some with ; in particular, . The previous expression now becomes

Using Mertens’ theorem, we thus conclude an exact formula for , and similarly for :

Proposition 1 (Exact formula) We have

where

Similarly we have

where and are defined similarly to and by replacing all occurrences of with .

These formulae are unwieldy. However if we make some monotonicity hypotheses, namely that is positive, is negative, and is positive on , then we can get some good estimates on the (which are now non-negative functions) and hence on . Namely, if is negative but increasing then we have

for and , which implies that

for any . A similar argument in fact gives

for any . Iterating this we conclude that

and similarly

From Cauchy-Schwarz we thus have

Observe from the binomial formula that of the pairs with , of them have even, and of them have odd. We thus have

We have thus established the upper bound

where

By symmetry we may factorise

The expression is explicitly computable for any given . Following the recent preprint of Pintz, one can get a slightly looser, but cleaner, bound by using the upper bound

and so

Note that

and hence

where

In practice we expect the term to dominate, thus we have the heuristic approximation

Now we turn to the estimation of . We have an analogue of (8), namely

But we have an improvment in the lower bound in the case, because in this case we have

From the positive decreasing nature of we see that and so is non-negative and can thus be ignored for the purposes of lower bounds. (There are similar improvements available for higher but this seems to only give negligible improvements and will not be pursued here.) Thus we obtain

where

Estimating similarly to we conclude that

where

By (9), (10), we see that the condition (1) is implied by

By Theorem 14 and Lemma 15 of this previous post, we may take the ratio to be arbitrarily close to . We conclude the following theorem.

Theorem 2 Let and be such that holds. Let be an integer, define

and

and suppose that

Then holds.

As noted earlier, we heuristically have

and is negligible. This constraint is a bit better than the previous condition, in which was not present and was replaced by a quantity roughly of the form .

First, some (rough) motivational background, omitting all the number-theoretic details and focusing on the combinatorics. (But readers who just want to see the combinatorial problem can skip the motivation and jump ahead to Lemma 5.) As part of the Polymath8 project we are trying to establish a certain estimate called for as wide a range of as possible. Currently the best result we have is:

Theorem 1 (Zhang’s theorem, numerically optimised) holds whenever .

Enlarging this region would lead to a better value of certain parameters , which in turn control the best bound on asymptotic gaps between consecutive primes. See this previous post for more discussion of this. At present, the best value of is applied by taking sufficiently close to , so improving Theorem 1 in the neighbourhood of this value is particularly desirable.

I’ll state exactly what is below the fold. For now, suffice to say that it involves a certain number-theoretic function, the von Mangoldt function . To prove the theorem, the first step is to use a certain identity (the Heath-Brown identity) to decompose into a lot of pieces, which take the form

for some bounded (in Zhang’s paper never exceeds ) and various weights supported at various scales that multiply up to approximately :

We can write , thus ignoring negligible errors, are non-negative real numbers that add up to :

A key technical feature of the Heath-Brown identity is that the weight associated to sufficiently large values of (e.g. ) are “smooth” in a certain sense; this will be detailed below the fold.

The operation is Dirichlet convolution, which is commutative and associative. We can thus regroup the convolution (1) in a number of ways. For instance, given any partition into disjoint sets , we can rewrite (1) as

where is the convolution of those with , and similarly for .

Zhang’s argument splits into two major pieces, in which certain classes of (1) are established. Cheating a little bit, the following three results are established:

Theorem 2 (Type 0 estimate, informal version) The term (1) gives an acceptable contribution to whenever

for some .

Theorem 3 (Type I/II estimate, informal version) The term (1) gives an acceptable contribution to whenever one can find a partition such that

where is a quantity such that

Theorem 4 (Type III estimate, informal version) The term (1) gives an acceptable contribution to whenever one can find with distinct with

and

The above assertions are oversimplifications; there are some additional minor smallness hypotheses on that are needed but at the current (small) values of under consideration they are not relevant and so will be omitted.

The deduction of Theorem 1 from Theorems 2, 3, 4 is then accomplished from the following, purely combinatorial, lemma:

Lemma 5 (Subset sum lemma) Let be such that

Let be non-negative reals such that

Then at least one of the following statements hold:

• (Type 0) There is such that .
• (Type I/II) There is a partition such that

  

  where is a quantity such that

  

• (Type III) One can find distinct with

  

  and

  

The purely combinatorial question is whether the hypothesis (2) can be relaxed here to a weaker condition. This would allow us to improve the ranges for Theorem 1 (and hence for the values of and alluded to earlier) without needing further improvement on Theorems 2, 3, 4 (although such improvement is also going to be a focus of Polymath8 investigations in the future).

Let us review how this lemma is currently proven. The key sublemma is the following:

Lemma 6 Let , and let be non-negative numbers summing to . Then one of the following three statements hold:

• (Type 0) There is a with .
• (Type I/II) There is a partition such that

  

• (Type III) There exist distinct with and .

Proof: Suppose Type I/II never occurs, then every partial sum is either “small” in the sense that it is less than or equal to , or “large” in the sense that it is greater than or equal to , since otherwise we would be in the Type I/II case either with as is and the complement of , or vice versa.

Call a summand “powerless” if it cannot be used to turn a small partial sum into a large partial sum, thus there are no such that is small and is large. We then split where are the powerless elements and are the powerful elements.

By induction we see that if and is small, then is also small. Thus every sum of powerful summand is either less than or larger than . Since a powerful element must be able to convert a small sum to a large sum (in fact it must be able to convert a small sum of powerful summands to a large sum, by stripping out the powerless summands), we conclude that every powerful element has size greater than . We may assume we are not in Type 0, then every powerful summand is at least and at most . In particular, there have to be at least three powerful summands, otherwise cannot be as large as . As , we have , and we conclude that the sum of any two powerful summands is large (which, incidentally, shows that there are exactly three powerful summands). Taking to be three powerful summands in increasing order we land in Type III.

Now we see how Lemma 6 implies Lemma 5. Let be as in Lemma 5. We take almost as large as we can for the Type I/II case, thus we set

for some sufficiently small . We observe from (2) that we certainly have

and

with plenty of room to spare. We then apply Lemma 6. The Type 0 case of that lemma then implies the Type 0 case of Lemma 5, while the Type I/II case of Lemma 6 also implies the Type I/II case of Lemma 5. Finally, suppose that we are in the Type III case of Lemma 6. Since

we thus have

and so we will be done if

Inserting (3) and taking small enough, it suffices to verify that

but after some computation this is equivalent to (2).

It seems that there is some slack in this computation; some of the conclusions of the Type III case of Lemma 5, in particular, ended up being “wasted”, and it is possible that one did not fully exploit all the partial sums that could be used to create a Type I/II situation. So there may be a way to make improvements through purely combinatorial arguments. (UPDATE: As it turns out, this is sadly not the case: consderation of the case when , , and shows that one cannot obtain any further improvement without actually improving the Type I/II and Type III analysis.)

A technical remark: for the application to Theorem 1, it is possible to enforce a bound on the number of summands in Lemma 5. More precisely, we may assume that is an even number of size at most for any natural number we please, at the cost of adding the additioal constraint to the Type III conclusion. Since is already at least , which is at least , one can safely take , so can be taken to be an even number of size at most , which in principle makes the problem of optimising Lemma 5 a fixed linear programming problem. (Zhang takes , but this appears to be overkill. On the other hand, does not appear to be a parameter that overly influences the final numerical bounds.)

Below the fold I give the number-theoretic details of the combinatorial aspects of Zhang’s argument that correspond to the combinatorial problem described above.

— 1. Coefficient sequences —

We now give some number-theoretic background material that will serve two purposes. The most immediate purpose is to enable one to understand the precise statement of Theorems 1, 2, 3, 4, as well as the deduction of the first theorem from the other three. A secondary purpose is to establish some reference material which will be used in subsequent posts on the Type I/II and Type III analysis in Zhang’s arguments.

As in previous posts, we let be an asymptotic parameter tending to infinity and define the usual asymptotic notation relative to this parameter. It is also convenient to have a large fixed quantity to be chosen later, in order to achieve a fine localisation of scales.

It will be convenient to set up some notation regarding certain types of sequences, abstracting some axioms appearing in the work of Bombieri-Friedlander-Iwaniec and Zhang:

Definition 7 A coefficient sequence is a finitely supported sequence that obeys the bounds

for all , where is the divisor function. (In particular, any sequence that is pointwise dominated by a coefficient sequence is again a coefficient sequence.)

• (i) If is a coefficient sequence and is a primitive residue class, the (signed) discrepancy of in the sequence is defined to be the quantity

  

  Note that this expression is linear in , so in particular we have the triangle inequality

  

• (ii) A coefficient sequence is said to be at scale for some if it is supported on an interval of the form .
• (iii) A coefficient sequence at scale is said to obey the Siegel-Walfisz theorem if one has

  

  for any , any fixed , and any primitive residue class .

• (iv) A coefficient sequence at scale is said to obey the Elliott-Halberstam conjecture up to scale for some if one has

  

  for any fixed . Thus for instance the Siegel-Walfisz theorem implies the Elliott-Halberstam conjecture up to scale for any fixed .

• (v) A coefficient sequence at scale is said to be smooth if it takes the form for some smooth function supported on obeying the derivative bounds

  

  for all fixed (note that the implied constant in the notation may depend on .

To control error terms, we will frequently use the following crude bounds:

Lemma 8 (Crude estimates) Let be a coefficient sequence, let be fixed, and let . Then we have

and

more generally one has

for any congruence class (not necessarily primitive). In particular, if is at scale for some , one has

and

and hence from (5) we have the crude discrepancy bound

(In particular, the Siegel-Walfisz estimate (7) is trivial unless .) Finally, we have the crude bound

for all .

Proof: To prove (10) it suffices from (4) to show that

but this follows from e.g. Proposition 5 of this previous post. From (10) we also deduce (11). To show (12), it suffices to show that

(i.e. a Brun-Titchmarsh type inequality for powers of the divisor function); this estimate follows from this paper of Shiu or this paper of Barban-Vehov, and can also be proven using the methods in this previous blog post. (The factor of is needed to account for the possibility that is not primitive, while the term accounts for the possibility that is as large as .) Finally, (16) follows from the standard divisor bound; see this previous post.

As a general rule, we may freely use (16) when we are expecting a net power savings to come from another part of the analysis, and we may freely use the other estimates in Lemma 8 when we have a net super-logarithmic savings from elsewhere in the analysis. When we have neither a net power savings or a net super-logarithmic savings from other sources then we usually cannot afford to use any of the bounds in Lemma 8.

The concept of a coefficient sequence is stable under the operation of Dirichlet convolution operation

Lemma 9 Let be coefficient sequences. Then:

• (i) is also a coefficient sequence.
• (ii) If are coefficient sequences at scales respectively, then is a coefficient sequence at scale .
• (iii) If are coefficient sequences at scales respectively, with for some fixed , and obeys a Siegel-Walfisz theorem, then also obeys a Siegel-Walfisz theorem (at scale ).
• (iv) If are coefficient sequences at scales respectively, with for some fixed , and obeys the Elliott-Halberstam conjecture up to some scale , then (viewed as a coefficient sequence at scale ) also obeys the Elliott-Halberstam conjecture up to scale .
• (v) If is the logarithm function, then is a coefficient sequence. Furthermore, if is smooth at some scale , then is smooth at that scale also.

Proof: To verify (i), observe from (4) that for any one has

so that is again a coefficient sequence. The claim (ii) then follows by considering the support of .

Now we verify (iii). We need to show that

for any and any residue class . By restricting to integers coprime to (and noting that the restricted version of still obeys the Siegel-Walfisz property if one divides out by a suitable power of ) we may assume that are supported on integers coprime to , at which point we may drop the constraint. As noted in Lemma 8 we may also assume that .

We now have

and

and by the triangle inequality so we may upper bound by

Applying (7) for and (14) for , we may bound this by

for any fixed , which is acceptable using the bound on .

Now we verify (iv), which is similar to (iii). Arguing as before, we have the inequality

By (13) and the Elliott-Halberstam hypothesis for we thus have

for any fixed and some fixed . However from two applications of Lemma 8 we also have

and the claim now follows from the Cauchy-Schwarz inequality.

Finally, (v) is an easy consequence of the product rule and is left to the reader.

We also have some basic sequences that obey the Siegel-Walfisz property:

Lemma 10 Let be a smooth coefficient sequence at some scale with for some fixed . Then

• (i) obeys the Siegel-Walfisz theorem.
• (ii) In fact, obeys the Elliott-Halberstam conjecture up to scale for any fixed .
• (iii) obeys the Siegel-Walfisz theorem, where is the Möbius function.

Note that some lower bound on is necessary here, since one cannot hope for a sequence to be equidistributed to the extent predicted by the Siegel-Walfisz theorem (7) if the sequence is only supported on a logarithmic range!

Proof: We first prove (i). Our task is to show that

for any and any residue class . By applying the Möbius inversion formula

and the triangle inequality (6) we thus see that it suffices to show that

for any . We may assume that is coprime to as the left-hand side vanishes otherwise. Our task is now to show that

For this it will suffice to establish the claim

for any residue class (not necessarily primitive). Note that we may assume that as the claim follows from (14) otherwise. We use the Fourier expansion

where for or (by abuse of notation) for . We can thus write the left-hand side of (17) as

The term here is the first term on the right-hand side of (17). Thus it will suffice to show that

for all non-zero . But if we write , then by the Poisson summation formula the left-hand side is equal to

where is the Fourier transform of . From the smoothness bounds (9) and integration by parts we have

for any fixed , so for large enough (actually is enough) we obtain the claim thanks to the lower bound and the upper bound . We remark that the same argument (now with as large as needed) in fact shows the much stronger bound

for any fixed and any , which also gives (ii).

To prove (iii), we can argue similarly to before and reduce to showing that

for any , but this follows from the Siegel-Walfisz theorem for the Möbius function (which can be deduced from the more usual Siegel-Walfisz theorem for the von Mangoldt function, or else proven by essentially the same method) and summation by parts (if one wishes, one can also reduce to the case of primitive residue classes using the multiplicativity of ).

We remark that the Siegel-Walfisz theorem for the Möbius function is ineffective, although in practice one can obtain effective substitutes for this theorem that can make applications (such as the one for bounded prime gaps) effective, see e.g. the last section of this article of Pintz for further discussion. One amusing remark in this regard is that if there do happen to be infinitely many Siegel zeroes, then an old result of Heath-Brown shows that there are infinitely mnay twin primes already, so the ineffective case of Siegel-Walfisz’s theorem is in some sense the best case scenario for us!

Lemmas 9 and 10 combine to give plenty of coefficient sequences obeying the Siegel-Walfisz property. This will become useful when the time comes to deduce Theorem 1 from (the precise versions of) Theorems 2, 3, 4.

— 2. Congruence class systems —

The Elliott-Halberstam conjecture on a sequence at scale requires taking a supremum over all primitive residue classes. At present, we do not know how to achieve such a strong claim for non-smooth arithmetic sequences such as or once the modulus goes beyond the level , even if one restricts to smooth moduli. However, Zhang’s work (as well as some of the precursor work of Bombieri, Fouvry, Friedlander, and Iwaniec) allow one to get a restricted version of the Elliott-Halberstam conjecture if one restricts the congruences one is permitted to work with.

Much as with the coefficient sequence axioms, it is convenient to abstract the axioms that a given system of congruence classes will be obeying. For any set , let denote the set of squarefree natural numbers whose prime divisors lie in .

Definition 11 Let . A congruence class system on is a collection of sets residue classes for each obeying the following axioms:

• (i) (Primitivity) For each , is a subset of .
• (ii) (Chinese remainder theorem) For any coprime , we have , using the canonical identification between and .
• (iii) (Uniform bound) There is a fixed such that for all primes .

If for all , thus , we say that is a singleton congruence class system.

For any integer , let denote the multiplicity function

or equivalently

A congruence class system is said to have controlled multiplicity if for any congruence class with , we have

for any fixed .

Note from Axiom (ii) that a congruence class system can be specified through its values at primes . A simple example of a singleton congruence class system with controlled multiplicity is a fixed congruence class for some fixed non-zero , with avoiding all the prime divisors of . This is a special case of the following more general fact:

Lemma 12 Let be a fixed admissible -tuple for some fixed , and let . For any modulus , set

where . Then for any , is a congruence class system on with controlled multiplicity.

Similarly, if is such that is coprime to for all and , then the system defined by setting when is coprime to , and when divides , with then defined for arbitrary by the Chinese remainder theorem, is also a congruence system on with controlled multiplicity.

Proof: We just prove the first claim, as the second is similar. Axioms (i)-(iii) are obvious; it remains only to verify the controlled multiplicity. Let be a congruence class with .

We can split

and hence

so it suffices to show that

for each . Writing , this becomes

by dividing through by (using the factor to absorb the losses) we may assume that is primitive. The claim then follows Lemma 8.

The more precise statement of Theorem 1 is now as follows:

Theorem 13 (Zhang’s theorem, numerically optimised) Let be fixed parameters such that

Let , and let be a congruence class system with controlled multiplicity. Then

for any fixed , where is the von Mangoldt function.

For the application to prime gaps we only need to apply Theorem 13 to the congruence class system associated to a fixed -tuple through Lemma 12, but one could imagine that this theorem could have future application with some other congruence class systems.

One advantage of the abstract formulation of a congruence class systems is that we get a cheap reduction to the singleton case:

Proposition 14 (Reduction to singletons) In order to prove Theorem 13, it suffices to do so for good singleton class congruence systems.

Proof: Let be a good congruence class system. By removing all primes with from we may assume without loss of generality that for all and some fixed .

We use the probabilistic method. Construct a random singleton congruence class system by selecting uniformly at random from independently for all , and then set and extend by the Chinese remainder theorem. Writing , we observe that the property that enjoys of being a congruence class system of controlled multiplicity is inherited by . From hypothesis we then have that

for any fixed ; taking expectations we conclude that

On the other hand, from Lemma 8 we have

and the claim then follows from Cauchy-Schwarz.

This reduction is not essential to Zhang’s argument (indeed, it is not used in Zhang’s paper), but it does allow for a slightly simpler notation (since the summation over is eliminated).

We can now also state the precise versions of Theorems 2, 3, 4 that we need:

Theorem 15 (Type 0 estimate, precise version) Let be fixed, and let be coefficient sequences at scales respectively with smooth,

and

for some fixed . Then obeys the Elliott-Halberstam conjecture up to scale . In particular, for any and any singleton congruence class system we have

Proof: This is immediate from Lemma 10(ii) and Lemma 7(iv).

Theorem 16 (Type I/II estimate, precise version) Let be fixed quantities such that

and

and let be coefficient sequences at scales respectively with

and

with obeying a Siegel-Walfisz theorem. Then for any and any singleton congruence class system with controlled multiplicity we have

The condition (22) is dominated by (20) and can thus be ignored, at least at our current numerical ranges of parameters. We remark that this theorem is the only theorem that actually uses the controlled multiplicity hypothesis.

Theorem 17 (Type III estimate, precise version) Let be fixed quantities. Let be scales obeying the relations

and

for some fixed . Let be coefficient sequences at scales respectively, with smooth. Then for any , and any be a singleton congruence class system we have

for any fixed .

NOTE: actually there should be some additional constraint on in the hypotheses of Theorem 17 similar to (22) that we have not fully identified yet, but again it will be dominated by (20) with plenty of room to spare and so can be ignored for the purposes of this post. (I will add in this hypothesis soon, though, after we have gone through the Type III analysis in detail.)

As we saw, Theorem 15 is easy to establish. On the other hand, Theorem 16 and Theorem 17 are far deeper and will be the subject of future blog posts. We will not discuss them further here, but now turn to the question of how to deduce Theorem 13 from Theorems 15, 16, 17 using Lemma 5.

— 3. Decomposing the von Mangoldt function —

The basic strategy is to decompose the von Mangoldt function as a combination of Dirichlet convolutions of other functions such as , the constant function , and the logarithmic function . The simplest identity of this form is

but this is unsuitable for our purposes because when we localise to scale , the factor could also be localised at scales as large as , and our understanding of the equidistribution properties of the Möbius function is basically no better than that of the von Mangoldt function, so we have not gained anything with this decomposition.

To get around this we need to find decompositions that don’t let rough functions such as the Möbius function get up to scales anywhere close to . One promising identity in this regard is Linnik’s identity, which takes the form

where is the restriction of the constant function to numbers larger than ; this is just the coefficient version of the formal geometric series identity

There are no Möbius functions in sight on the right-hand side, which is promising. However, Linnik’s identity has an unbounded number of terms, which render it unsuitable for our argument (we have various factors of and whose exponent would become unbounded if we had to deal with Dirichlet convolutions of unbounded length, which would swamp any gain of that we are trying to detect). So we will rely on a truncated variant of Linnik’s identity, known as the Heath-Brown identity.

We will need a fixed positive natural number (Zhang takes ; the precise choice here does not seem to be terribly important). From the identities and , where is the Dirichlet convolution identity, we see that we can write as a -fold convolution

where denotes the convolution of copies of .

As with (23), the identity (24) is not directly useful for our strategy because one of the factors can still get as large as the scale that we are studying at. However, as Heath-Brown observed, we can manipulate (24) into a useful form by truncating the Möbius function . More precisely, we split the Möbius function as

where is the Möbius function restricted to the interval , and is the Möbius function restricted to the interval . The reason for this splitting is that the -fold Möbius convolution vanishes on , and in particular we have

on . Expanding out and using the binomial formula, we conclude that

on .

By (24), the term of (25) is just . For all the other terms, we can cancel against using the Möbius inversion formula to conclude the Heath-Brown identity

on . Now we see that each of the Möbius factors cannot reach scales much larger than , although the factors may collectively still get close to when is close to . From the triangle inequality and Proposition, we thus see that to establish Theorem 13, it suffices to establish the bounds

whenever are fixed and obey (20), , , is fixed, and is a singleton congruence class system with controlled multiplicity.

This is now looking closer to the type of estimates that can be handled by Theorems 15, 16, 17, but there are still some technical issues to resolve, namely the presence of the cutoff and also the fact that the functions , , are not localised to any given scale, but are instead spread out at across many scales. This however can be dealt with by a routine dyadic decomposition (which, in harmonic analysis, is sometimes referred to as Littlewood-Paley decomposition, at least when applied in the frequency domain), though here instead of using the usual dyadic range of scales, one uses instead a sub-dyadic range to eliminate edge effects. (This trick dates back at least to the work of Fouvry; thanks to Emmanuel Kowalski for this reference.)

More precisely, let be a large fixed number to be chosen later, and let be a smooth function supported on that equals on and obeys the derivative estimates

for any fixed (note that the implied constant here can depend on ). We then have a smooth partition of unity

indexed by the multiplicative semigroup for any natural number , where

We can thus decompose

and similarly

and

For , the expression

can thus be split as

which we can rewrite as

where is a variant of .

Observe that the summand vanishes unless

and

and the factor can be eliminated except in the boundary cases when

or

Let us deal with the contribution of the boundary cases to (26). If we let be the sum of all the boundary summands, then we have

From Lemma 8 we have that

and

for any fixed and any . On the other hand, is supported on two intervals of length . Thus by Cauchy-Schwarz one has

and

for any fixed . We thus have

and hence by Lemma 8 again

which is acceptable for (26) if we take large enough. Thus it suffices to deal with the contribution of the interior summands, in which the cutoff may be dropped. There are such summands, and the factor is bounded by , so it will suffice to show that

whenever and are such that each takes the form , , or . In particular, from Lemmas 9, (10) we observe the following facts for any and :

• (i) Each is a coefficient sequence at scale . More generally the convolution of the for is a coefficient sequence at scale .
• (ii) If for some fixed , then is smooth.
• (iii) If for some fixed , then obeys a Siegel-Walfisz theorem. More generally, obeys a Siegel-Walfisz theorem if for some fixed .
• (iv) .

Now we can prove (27). We can write for , where the are non-negative reals that sum to . We apply Lemma 5 and conclude that the obey one of the three conclusions (Type 0), (Type I/II), (Type III) of that lemma. Furthermore, an inspection of the proof of that lemma (and the fact that the hypothesis (5) shows that any strict inequalities such as can in fact be strengthened infinitesimally to for some fixed . Furthermore, in the Type III case, an inspection of Lemma 6 reveals that we have an additional lower bound available, which in particular implies that for some fixed if is large enough ( will certainly do).

In the Type O case, we can write in a form in which Theorem 15 applies. Similarly, in the Type I/II case we can write in a form in which Theorem 16 applies, and in the Type III case we can write in a form in which Theorem 17 applies. Thus in all cases we can establish (27), and Theorem 13 follows.

We will continue the notation from the previous post, including the concept of an admissible tuple, the use of an asymptotic parameter going to infinity, and a quantity depending on that goes to infinity sufficiently slowly with , and (the -trick).

The objective of this portion of the Polymath8 project is to make as efficient as possible the connection between two types of results, which we call and . Let us first state , which has an integer parameter :

Conjecture 1 () Let be a fixed admissible -tuple. Then there are infinitely many translates of which contain at least two primes.

Zhang was the first to prove a result of this type with . Since then the value of has been lowered substantially; at this time of writing, the current record is .

There are two basic ways known currently to attain this conjecture. The first is to use the Elliott-Halberstam conjecture for some :

Conjecture 2 () One has

for all fixed . Here we use the abbreviation for .

Here of course is the von Mangoldt function and the Euler totient function. It is conjectured that holds for all , but this is currently only known for , an important result known as the Bombieri-Vinogradov theorem.

In a breakthrough paper, Goldston, Yildirim, and Pintz established an implication of the form

for any , where depends on . This deduction was very recently optimised by Farkas, Pintz, and Revesz and also independently in the comments to the previous blog post, leading to the following implication:

Theorem 3 (EH implies DHL) Let be a real number, and let be an integer obeying the inequality

where is the first positive zero of the Bessel function . Then implies .

Note that the right-hand side of (2) is larger than , but tends asymptotically to as . We give an alternate proof of Theorem 3 below the fold.

Implications of the form Theorem 3 were modified by Motohashi and Pintz, which in our notation replaces by an easier conjecture for some and , at the cost of degrading the sufficient condition (2) slightly. In our notation, this conjecture takes the following form for each choice of parameters :

Conjecture 4 () Let be a fixed -tuple (not necessarily admissible) for some fixed , and let be a primitive residue class. Then

for any fixed , where , are the square-free integers whose prime factors lie in , and is the quantity

and is the set of congruence classes

and is the polynomial

This is a weakened version of the Elliott-Halberstam conjecture:

Proposition 5 (EH implies MPZ) Let and . Then implies for any . (In abbreviated form: implies .)

In particular, since is conjecturally true for all , we conjecture to be true for all and .

Proof: Define

then the hypothesis (applied to and and then subtracting) tells us that

for any fixed . From the Chinese remainder theorem and the Siegel-Walfisz theorem we have

for any coprime to (and in particular for ). Since , where is the number of prime divisors of , we can thus bound the left-hand side of (3) by

The contribution of the second term is by standard estimates (see Proposition 8 below). Using the very crude bound

and standard estimates we also have

and the claim now follows from the Cauchy-Schwarz inequality.

In practice, the conjecture is easier to prove than due to the restriction of the residue classes to , and also the restriction of the modulus to -smooth numbers. Zhang proved for any . More recently, our Polymath8 group has analysed Zhang’s argument (using in part a corrected version of the analysis of a recent preprint of Pintz) to obtain whenever are such that

The work of Motohashi and Pintz, and later Zhang, implicitly describe arguments that allow one to deduce from provided that is sufficiently large depending on . The best implication of this sort that we have been able to verify thus far is the following result, established in the previous post:

Theorem 6 (MPZ implies DHL) Let , , and let be an integer obeying the constraint

where is the quantity

Then implies .

This complicated version of is roughly of size . It is unlikely to be optimal; the work of Motohashi-Pintz and Pintz suggests that it can essentially be improved to , but currently we are unable to verify this claim. One of the aims of this post is to encourage further discussion as to how to improve the term in results such as Theorem 6.

We remark that as (5) is an open condition, it is unaffected by infinitesimal modifications to , and so we do not ascribe much importance to such modifications (e.g. replacing by for some arbitrarily small ).

The known deductions of from claims such as or rely on the following elementary observation of Goldston, Pintz, and Yildirim (essentially a weighted pigeonhole principle), which we have placed in “-tricked form”:

Lemma 7 (Criterion for DHL) Let . Suppose that for each fixed admissible -tuple and each congruence class such that is coprime to for all , one can find a non-negative weight function , fixed quantities , a quantity , and a fixed positive power of such that one has the upper bound

the lower bound

for all , and the key inequality

holds. Then holds. Here is defined to equal when is prime and otherwise.

Proof: Consider the quantity

By (6), (7), this quantity is at least

By (8), this expression is positive for all sufficiently large . On the other hand, (9) can only be positive if at least one summand is positive, which only can happen when contains at least two primes for some with . Letting we obtain as claimed.

In practice, the quantity (referred to as the sieve level) is a power of such as or , and reflects the strength of the distribution hypothesis or that is available; the quantity will also be a key parameter in the definition of the sieve weight . The factor reflects the order of magnitude of the expected density of in the residue class ; it could be absorbed into the sieve weight by dividing that weight by , but it is convenient to not enforce such a normalisation so as not to clutter up the formulae. In practice, will some combination of and .

Once one has decided to rely on Lemma 7, the next main task is to select a good weight for which the ratio is as small as possible (and for which the sieve level is as large as possible. To ensure non-negativity, we use the Selberg sieve

where takes the form

for some weights vanishing for that are to be chosen, where is an interval and is the polynomial . If the distribution hypothesis is , one takes and ; if the distribution hypothesis is instead , one takes and .

One has a useful amount of flexibility in selecting the weights for the Selberg sieve. The original work of Goldston, Pintz, and Yildirim, as well as the subsequent paper of Zhang, the choice

is used for some additional parameter to be optimised over. More generally, one can take

for some suitable (in particular, sufficiently smooth) cutoff function . We will refer to this choice of sieve weights as the “analytic Selberg sieve”; this is the choice used in the analysis in the previous post.

However, there is a slight variant choice of sieve weights that one can use, which I will call the “elementary Selberg sieve”, and it takes the form

for a sufficiently smooth function , where

for is a -variant of the Euler totient function, and

for is a -variant of the function . (The derivative on the cutoff is convenient for computations, as will be made clearer later in this post.) This choice of weights may seem somewhat arbitrary, but it arises naturally when considering how to optimise the quadratic form

(which arises naturally in the estimation of in (6)) subject to a fixed value of (which morally is associated to the estimation of in (7)); this is discussed in any sieve theory text as part of the general theory of the Selberg sieve, e.g. Friedlander-Iwaniec.

The use of the elementary Selberg sieve for the bounded prime gaps problem was studied by Motohashi and Pintz. Their arguments give an alternate derivation of from for sufficiently large, although unfortunately we were not able to confirm some of their calculations regarding the precise dependence of on , and in particular we have not yet been able to improve upon the specific criterion in Theorem 6 using the elementary sieve. However it is quite plausible that such improvements could become available with additional arguments.

Below the fold we describe how the elementary Selberg sieve can be used to reprove Theorem 3, and discuss how they could potentially be used to improve upon Theorem 6. (But the elementary Selberg sieve and the analytic Selberg sieve are in any event closely related; see the appendix of this paper of mine with Ben Green for some further discussion.) For the purposes of polymath8, either developing the elementary Selberg sieve or continuing the analysis of the analytic Selberg sieve from the previous post would be a relevant topic of conversation in the comments to this post.

— 1. Sums of multiplicative functions —

In this section we review a standard estimate on a sum of multiplicative functions. We fix an interval . For any positive integer , we say that a multiplicative function has dimension if one has the asymptotic

for all ; in particular (since as ) we see that is non-negative on for large enough. Thus for instance

has dimension one, the divisor function

has dimension two, and the functions

and

defined in the introduction have dimension . Dimension interacts well with multiplication; the product of a -dimensional multiplicative function and a -dimensional multiplicative function is clearly a -multiplicative function.

We have the following basic asymptotic in the untruncated case :

Lemma 8 (Untruncated asymptotic) Let Let be a fixed positive integer, and let be a multiplicative function of dimension . Then for any fixed compactly supported, Riemann-integrable function , and any that goes to infinity as , one has

Proof: By approximating from above and below by smooth compactly supported functions we see that we may assume without loss of generality that is smooth and compactly supported. But then the claim follows from Proposition 10 of the previous post.

We remark that Proposition 10 of the previous post also gives asymptotics for a number of other sums of multiplicative functions, but one (small) advantage of the elementary Selberg sieve is that these (slightly) more complicated asymptotics are not needed. The generalisation in Lemma 8 from smooth to Riemann integrable implies in particular that

and conversely Lemma 8 can be easily deduced from (12) by another approximation argument (using piecewise constant functions instead of smooth functions). We also make the trivial remark that if is non-negative and is any subset of , then we have the upper bound

for any non-negative Riemann integrable .

Actually, (12) can be derived by purely elementary means (without the need to explicitly work with asymptotics of zeta functions as was done in the previous post) by an induction on the dimension as follows. In the dimension zero case we have the Euler product

and hence

which gives (12) in the case.

Now suppose that has dimension . In this case we write

where is a multiplicative function with

for all and , and for and . Then the left-hand side of (12) can be rearranged as

Elementary sieving gives

and hence by summation by parts

Meanwhile we have

and so

From these estimates one easily obtains (12) for .

Now suppose that and that the claim has been proven inductively for . We again may decompose (14), but now has dimension instead of dimension zero. Arguing as before, we can write the left-hand side of (12) as

The contribution of the and error terms are acceptable by induction hypothesis, and the main term is also acceptable from induction hypothesis and summation by parts, giving the claim.

— 2. Untruncated implication —

We first reprove Theorem 3. The key calculations for and are as follows:

Lemma 9 (Untruncated sieve bounds) Assume holds for some and some . Let be a smooth function that is supported on , let be a fixed admissible -tuple for some fixed , let be such that is coprime to for all , and let be the elementary Selberg sieve with weights (11) associated to the function , the sieve level and the untruncated interval . Then (6), (7) hold with

and

As computed in Theorem 14 of the previous post (and also in the recent preprint of Farkas, Pintz, and Revesz), the ratio

for non-zero can be made arbitrarily close to (the extremiser is not quite smooth at if one extends by zero for , but this can be easily dealt with by a standard regularisation argument), and Theorem 6 then follows from Lemma 7 (using the open nature of (2) to replace by for some small ).

It remains to prove Lemma 9. We begin with the proof of (6) (which will in fact be an asymptotic and not just an upper bound).

We expand the left-hand side of (6) as

The weights are only non-vanishing when . From the Chinese remainder theorem we then have

The contribution of the error term is

which we can upper bound by

Using (11) and Lemma 8 we have the crude upper bound

and hence by another application of Lemma 8 the previous expression may be upper bounded by , which is negligible by choice of . So we reduce to showing that

To proceed further we follow Selberg and observe the decomposition

for , which can be easily verified by working locally (when for some prime ) and then using multiplicativity. Using this identity we can diagonalise the left-hand side of (18) as

Now we use the form (11) of , which has been optimised specifically for ease of computing this expression. We can expand as

writing and , we can rewrite this as

which by Möbius inversion simplifies to

The left-hand side of (18) has now simplified to

By Lemma 8 and (15) we obtain (18) and hence (6) as required.

Now we turn to the more difficult lower bound (7) for a fixed (again we will be able to get an asymptotic here rather than just a lower bound). The left-hand side expands as

Again, may be restricted to at most , so that is at most . As before, the inner summand vanishes unless lies in one of the residue classes , where

and is the modified polynomial

The cardinality of is , where

We can thus estimate

where the error term is given by

By a modification of the proof of Proposition 5 we see that the hypothesis implies that

for any fixed and any multiplicative function of a fixed dimension . Using the bound (17) we can then conclude that the contribution of the error term to (7) is negligible. So (7) becomes

Analogously to (19) we have the decomposition

for , where is the function

We can thus diagonalise the left-hand side of (20) similarly to before as

We can expand as

writing and and noting that , we can rewrite this as

Observe that

so we can simplify the left-hand side of (20) as

where is the -dimensional multiplicative function

To control this sum, let us first pretend that the constraint was not present, thus suppose we had to estimate

By Proposition 8, the inner sum is equal to

which by the fundamental theorem of calculus simplifies to

We remark that the error term here is uniform in , because the translates are equicontinuous and thus uniformly Riemann integrable. We conclude that (23) is equal to

where the error term is again uniform in . By Proposition 8 and (16), this expression is equal to

as required.

Now we reinstate the condition , which turns out to be negligible thanks to the -trick. More precisely, we may use Möbius inversion to write

By the preceding discussion, the term of this sum is

Now we consider the terms, which are error terms. We may bound the total contribution of these terms in magnitude by

Arguing as before we have

and so the expression (25) becomes

where the implied constant in the notation can depend on . The square of this expression is then

The left-hand side of (20) is now expressed as the sum of the main term

and the error terms

for and

The main term has already been estimated as (24). From Proposition 8 we have

for , and so all of the error terms end up being , and (7) follows. This concludes the proof of Theorem 3.

— 3. Applying truncation —

Now we experiment with truncating the above argument to to obtain results of the shape of Theorem 6. Unfortunately thus far the results do not give very good explicit dependencies of on , but this may perhaps improve with further effort.

Assume holds for some and some . Let be a smooth function that is supported on , let be a fixed admissible -tuple for some fixed , let be such that is coprime to for all , and let be the elementary Selberg sieve with weights (11) associated to the function , the sieve level and the truncated interval . As before, we set

and seek the best values for for which we can establish the upper bound (6) and the lower bound (7). Arguing as in the previous section (using (13) to control error terms) we can reduce (6) to

If we crudely replace the truncated interval by the untruncated interval and apply Proposition 8 (or (13)) we may reuse the previous value

for here, but it is possible that we could do better than this.

Now we turn to (7). Arguing as in the previous section, we reduce to showing that

We can, if desired, discard the constraint here by arguing as in the previous section, leaving us with

Because we now seek a lower bound, we cannot simply pass to the untruncated interval (e.g. using (13)), and must proceed more carefully. A simple way to proceed (as was done by Motohashi and Pintz) is to just discard all less than , only retaining those in the region between and . The reason for doing this is that the parameter is then forced to be at most if one wants the summand to be non-zero, and so for the summation at least one can replace by without incurring any error. As in the previous section we then have

and so one can lower bound (26), up to negligible errors, by

If the truncated interval were replaced by the untruncated interval , then Proposition 8 would estimate this expression as

To deal with the truncated interval , we use a variant of the Buchstab identity, namely the easy inequality

for any non-negative function . Using this identity and Proposition 8, we find that we may lower bound (26), up to negligible errors, by

minus the sum

(The term comes from .)If is non-negative and non-increasing on , then we can upper bound

for , and so

On the other hand, from the prime number theorem we have

Putting all this together, we can thus obtain (7) with

where

and

Following Pintz, we may upper bound by and rescale to obtain

which we can crudely bound by

But of course we can also calculate and explicitly for any fixed choice of . We conclude the following variant of Theorem 6:

Theorem 10 (MPZ implies DHL) Let , , and let be an integer obeying the constraint

with given by (27), and some smooth supported on which is non-negative and non-increasing on . Then implies .

For large enough depending on the hypotheses in Theorem 10 can be verified (e.g. by setting for a reasonably large ) but the dependence is poor due to the localisation of the integral in the denominator to the narrow interval . But perhaps there is a way to not have such a strict localisation in these arguments.

Theorem 1 (Bounded gaps between primes) There exists a natural number such that there are infinitely many pairs of distinct primes with .

Zhang obtained the explicit value of for . A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some :

Theorem 2 () Let be an admissible -tuple, that is to say a tuple of distinct integers which avoids at least one residue class mod for every prime . Then there are infinitely many translates of that contain at least two primes.

Zhang obtained for . The current best value of is , as discussed in this previous blog post. To get from to Theorem 1, one has to exhibit an admissible -tuple of diameter at most . For instance, with , the narrowest admissible -tuple that we can construct has diameter , which explains the current world record. There is an active discussion on trying to improve the constructions of admissible tuples at this blog post; it is conceivable that some combination of computer search and clever combinatorial constructions could obtain slightly better values of for a given value of . The relationship between and is approximately of the form (and a classical estimate of Montgomery and Vaughan tells us that we cannot make much narrower than , see this previous post for some related discussion).

The second step in Zhang’s argument, which is somewhat less elementary (relying primarily on the sieve theory of Goldston, Yildirim, Pintz, and Motohashi), is to deduce from a certain conjecture for some . Here is one formulation of the conjecture, more or less as (implicitly) stated in Zhang’s paper:

Conjecture 3 () Let be an admissible tuple, let be an element of , let be a large parameter, and define

for any natural number , and

for any function . Let equal when is a prime , and otherwise. Then one has

for any fixed .

Note that this is slightly different from the formulation of in the previous post; I have reverted to Zhang’s formulation here as the primary purpose of this post is to read through Zhang’s paper. However, I have distinguished two separate parameters here instead of one, as it appears that there is some room to optimise by making these two parameters different.

In the previous post, I described how one can deduce from . Ignoring an exponentially small error , it turns out that one can deduce from whenever one can find a smooth function vanishing to order at least at such that

By selecting for a real parameter to optimise over, and ignoring the technical term alluded to previously (which is the only quantity here that depends on ), this gives from whenever

It may be possible to do better than this by choosing smarter choices for , or performing some sort of numerical calculus of variations or spectral theory; people interested in this topic are invited to discuss it in the previous post.

The final, and deepest, part of Zhang’s work is the following theorem (Theorem 2 from Zhang’s paper, whose proof occupies Sections 6-13 of that paper, and is about 32 pages long):

Theorem 4 (Zhang) is true for all .

The significance of the fraction is that Zhang’s argument proceeds for a general choice of , but ultimately the argument only closes if one has

(see page 53 of Zhang) which is equivalent to . Plugging in this choice of into (1) then gives with as stated previously.

Improving the value of in Theorem 4 would lead to improvements in and then as discussed above. The purpose of this reading seminar is then twofold:

1. Going through Zhang’s argument in order to improve the value of (perhaps by decreasing ); and
2. Gaining a more holistic understanding of Zhang’s argument (and perhaps to find some more “global” improvements to that argument), as well as related arguments such as the prior work of Bombieri, Fouvry, Friedlander, and Iwaniec that Zhang’s work is based on.

In addition to reading through Zhang’s paper, the following material is likely to be relevant:

• A recent blog post of Emmanuel Kowalski on the technical details of Zhang’s argument.
• Scanned notes from a talk related to the above blog post.
• A recent expository note by Fouvry, Kowalski, and Michel on a Friedlander-Iwaniec character sum relevant to this argument.
• This 1981 paper of Fouvry and Iwaniec which is the first result in the literature which is roughly of the type . (This paper seems to give a related result for and , if I read it correctly; I don’t yet understand what prevents this result or modifications thereof from being used in place of Theorem 4.)

I envisage a loose, unstructured format for the reading seminar. In the comments below, I am going to post my own impressions, questions, and remarks as I start going through the material, and I encourage other participants to do the same. The most obvious thing to do is to go through Zhang’s Sections 6-13 in linear order, but it may make sense for some participants to follow a different path. One obvious near-term goal is to carefully go through Zhang’s arguments for instead of , and record exactly how various exponents depend on , and what inequalities these parameters need to obey for the arguments to go through. It may be that this task can be done at a fairly superficial level without the need to carefully go through the analytic number theory estimates in that paper, though of course we should also be doing that as well. This may lead to some “cheap” optimisations of which can then propagate to improved bounds on and thanks to the other parts of the Polymath project.

Everyone is welcome to participate in this project (as per the usual polymath rules); however I would request that “meta” comments about the project that are not directly related to the task of reading Zhang’s paper and related works be placed instead on the polymath proposal page. (Similarly, comments regarding the optimisation of given and should be placed at this post, while comments on the optimisation of given should be given at this post. On the other hand, asking questions about Zhang’s paper, even (or especially!) “dumb” ones, would be very appropriate for this post and such questions are encouraged.

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).