This post is a continuation of the previous post on sieve theory, which is an ongoing part of the Polymath8 project. As the previous post was getting somewhat full, we are rolling the thread over to the current post.
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 analysis 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 :
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 system 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):
where is the first positive zero of the Bessel function , and are the quantities
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:
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:
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
holds. Then holds. Here is defined to equal when is prime and otherwise.
— 2. Pintz’s argument —
the elementary Selberg sieve defined by
are the multiplicative functions
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. (The constraints and are redundant given that , but we retain them for emphasis.) 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 .
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
Now let be a small fixed constant to be chosen later, and suppose the following claim holds:
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 post, we can write the left-hand side as
where is the -dimensional multiplicative function
Similarly for and . Multiplying, we obtain
On the other hand,
and so on dividing we have
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 , or if one runs out of such prime divisors. In the former case one ends up at least as large as , and in the latter case one has reached . Next, one multiplies to this the prime divisors of less than until one reaches or exceeds ; this is possible thanks to (11) in the former case and is automatic in the latter case, and gives a divisor between and as required.
Now we need to obtain a lower bound for (9). If we write
(the minus sign being to compensate for the non-positive nature of ) then we have
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
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
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
so we may bound this contribution up to negligible errors by
where is as in case (iii).
We introduce the quantities
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
We now place upper bounds on . In this previous post, the bounds
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