The twin prime conjecture, still unsolved, asserts that there are infinitely many primes such that is also prime. A more precise form of this conjecture is (a special case) of the Hardy-Littlewood prime tuples conjecture, which asserts that
as , where is the von Mangoldt function and is the twin prime constant
Because is almost entirely supported on the primes, it is not difficult to see that (1) implies the twin prime conjecture.
One can give a heuristic justification of the asymptotic (1) (and hence the twin prime conjecture) via sieve theoretic methods. Recall that the von Mangoldt function can be decomposed as a Dirichlet convolution
where is the Möbius function. Because of this, we can rewrite the left-hand side of (1) as
To compute this double sum, it is thus natural to consider sums such as
or (to simplify things by removing the logarithm)
The prime number theorem in arithmetic progressions suggests that one has an asymptotic of the form
where is the multiplicative function with for even and
for odd. Summing by parts, one then expects
and so we heuristically have
The Dirichlet series
has an Euler product factorisation
for ; comparing this with the Euler product factorisation
for the Riemann zeta function, and recalling that has a simple pole of residue at , we see that
has a simple zero at with first derivative
From this and standard multiplicative number theory manipulations, one can calculate the asymptotic
which concludes the heuristic justification of (1).
What prevents us from making the above heuristic argument rigorous, and thus proving (1) and the twin prime conjecture? Note that the variable in (2) ranges to be as large as . On the other hand, the prime number theorem in arithmetic progressions (3) is not expected to hold for anywhere that large (for instance, the left-hand side of (3) vanishes as soon as exceeds ). The best unconditional result known of the type (3) is the Siegel-Walfisz theorem, which allows to be as large as . Even the powerful generalised Riemann hypothesis (GRH) only lets one prove an estimate of the form (3) for up to about .
However, because of the averaging effect of the summation in in (2), we don’t need the asymptotic (3) to be true for all in a particular range; having it true for almost all in that range would suffice. Here the situation is much better; the celebrated Bombieri-Vinogradov theorem (sometimes known as “GRH on the average”) implies, roughly speaking, that the approximation (3) is valid for almost all for any fixed . While this is not enough to control (2) or (1), the Bombieri-Vinogradov theorem can at least be used to control variants of (1) such as
for various sieve weights whose associated divisor function is supposed to approximate the von Mangoldt function , although that theorem only lets one do this when the weights are supported on the range . This is still enough to obtain some partial results towards (1); for instance, by selecting weights according to the Selberg sieve, one can use the Bombieri-Vinogradov theorem to establish the upper bound
which is off from (1) by a factor of about . See for instance this blog post for details.
It has been difficult to improve upon the Bombieri-Vinogradov theorem in its full generality, although there are various improvements to certain restricted versions of the Bombieri-Vinogradov theorem, for instance in the famous work of Zhang on bounded gaps between primes. Nevertheless, it is believed that the Elliott-Halberstam conjecture (EH) holds, which roughly speaking would mean that (3) now holds for almost all for any fixed . (Unfortunately, the factor cannot be removed, as investigated in a series of papers by Friedlander, Granville, and also Hildebrand and Maier.) This comes tantalisingly close to having enough distribution to control all of (1). Unfortunately, it still falls short. Using this conjecture in place of the Bombieri-Vinogradov theorem leads to various improvements to sieve theoretic bounds; for instance, the factor of in (4) can now be improved to .
In two papers from the 1970s (which can be found online here and here respectively, the latter starting on page 255 of the pdf), Bombieri developed what is now known as the Bombieri asymptotic sieve to clarify the situation more precisely. First, he showed that on the Elliott-Halberstam conjecture, while one still could not establish the asymptotic (1), one could prove the generalised asymptotic
for all natural numbers , where the generalised von Mangoldt functions are defined by the formula
These functions behave like the von Mangoldt function, but are concentrated on -almost primes (numbers with at most prime factors) rather than primes. The right-hand side of (5) corresponds to what one would expect if one ran the same heuristics used to justify (1). Sadly, the case of (5), which is just (1), is just barely excluded from Bombieri’s analysis.
More generally, on the assumption of EH, the Bombieri asymptotic sieve provides the asymptotic
for any fixed and any tuple of natural numbers other than , where
is a further generalisation of the von Mangoldt function (now concentrated on -almost primes). By combining these asymptotics with some elementary identities involving the , together with the Weierstrass approximation theorem, Bombieri was able to control a wide family of sums including (1), except for one undetermined scalar . Namely, he was able to show (again on EH) that for any fixed and any continuous function on the simplex that had suitable vanishing at the boundary, the sum
was equal to
when was odd and
when was even, where the integral on is with respect to the measure (this is Dirac measure in the case ). In particular, we have
and the twin prime conjecture would be proved if one could show that is bounded away from zero, while (1) is equivalent to the assertion that is equal to . Unfortunately, no additional bound beyond the inequalities provided by the Bombieri asymptotic sieve is known, even if one assumes all other major conjectures in number theory than the prime tuples conjecture and its variants (e.g. GRH, GEH, GUE, abc, Chowla, …).
To put it another way, the Bombieri asymptotic sieve is able (on EH) to compute asymptotics for sums
without needing to know the unknown scalar , when is a function supported on almost primes of the form
for and some fixed , with vanishing elsewhere and for some continuous (symmetric) functions obeying some vanishing at the boundary, so long as the parity condition
is obeyed (informally: gives the same weight to products of an odd number of primes as to products of an even number of primes, or to put it another way, is asymptotically orthogonal to the Möbius function ). But when violates the parity condition, the asymptotic involves the unknown . This scalar thus embodies the “parity problem” for the twin prime conjecture (discussed in these previous blog posts).
Because the obstruction to the parity problem is only one-dimensional (on EH), one can replace any parity-violating weight (such as ) with any other parity-violating weight and obtain a logically equivalent estimate. For instance, to prove the twin prime conjecture on EH, it would suffice to show that
for some fixed , or equivalently that there are solutions to the equation in primes with and . (In some cases, this sort of reduction can also be made using other sieves than the Bombieri asymptotic sieve, as was observed by Ng.) As another example, the Bombieri asymptotic sieve can be used to show that the asymptotic (1) is equivalent to the asymptotic
where is the set of numbers that are rough in the sense that they have no prime factors less than for some fixed (the function clearly correlates with and so must violate the parity condition). One can replace with similar sieve weights (e.g. a Selberg sieve) that concentrate on almost primes if desired.
As it turns out, if one is willing to strengthen the assumption of the Elliott-Halberstam (EH) conjecture to the assumption of the generalised Elliott-Halberstam (GEH) conjecture (as formulated for instance in Claim 2.6 of the Polymath8b paper), one can also swap the factor in the above asymptotics with other parity-violating weights and obtain a logically equivalent estimate, as the Bombieri asymptotic sieve also applies to weights such as under the assumption of GEH. For instance, on GEH one can use two such applications of the Bombieri asymptotic sieve to show that the twin prime conjecture would follow if one could show that there are solutions to the equation
in primes with and , for some . Similarly, on GEH the asymptotic (1) is equivalent to the asymptotic
for some fixed , and similarly with replaced by other sieves. This form of the quantitative twin primes conjecture is appealingly similar to the (special case)
of the Chowla conjecture, for which there has been some recent progress (discussed for instance in these recent posts). Informally, the Bombieri asymptotic sieve lets us (on GEH) view the twin prime conjecture as a sort of Chowla conjecture restricted to almost primes. Unfortunately, the recent progress on the Chowla conjecture relies heavily on the multiplicativity of at small primes, which is completely destroyed by inserting a weight such as , so this does not yet yield a viable path towards the twin prime conjecture even assuming GEH. Still, the similarity is striking, and one can hope that further ways to attack the Chowla conjecture may emerge that could impact the twin prime conjecture. (Alternatively, if one assumes a sufficiently optimistic version of the GEH, one could perhaps relax the notion of “almost prime” to the extent that one could start usefully using multiplicativity at smallish primes, though this seems rather wishful at present, particularly since the most optimistic versions of GEH are known to be false.)
The Bombieri asymptotic sieve is already well explained in the original two papers of Bombieri; there is also a slightly different treatment of the sieve by Friedlander and Iwaniec, as well as a simplified version in the book of Friedlander and Iwaniec (in which the distribution hypothesis is strengthened in order to shorten the arguments. I’ve decided though to write up my own notes on the sieve below the fold; this is primarily for my own benefit, but may be useful to some readers also. I largely follow the treatment of Bombieri, with the one idiosyncratic twist of replacing the usual “elementary” Selberg sieve with the “analytic” Selberg sieve used in particular in many of the breakthrough works in small gaps between primes; I prefer working with the latter due to its Fourier-analytic flavour.
— 1. Controlling generalised von Mangoldt sums —
To prove (5), we shall first generalise it, by replacing the sequence by a more general sequence obeying the following axioms:
These axioms are a little bit stronger than what is actually needed to make the Bombieri asymptotic sieve work, but we will not attempt to work with the weakest possible axioms here.
We introduce the function
which is analytic for ; in particular it can be evaluated at to yield
There are two model examples of data to keep in mind. The first, discussed in the introduction, is when , then and is as in the introduction; one of course needs EH to justify axiom (iv) in this case. The other is when , in which case and for all . We will later take advantage of the second example to avoid doing some (routine, but messy) main term computations.
The main result of this section is then
Theorem 1 Let be as above. Let be a tuple of natural numbers (independent of ) that is not equal to . Then one has the asymptotic
as , where .
Note that this recovers (5) (on EH) as a special case.
We now begin the proof of this theorem. Henceforth we allow implied constants in the or notation to depend on and .
It will be convenient to replace the range by a shorter range by the following standard localisation trick. Let be a large quantity depending on to be chosen later, and let denote the interval . We will show the estimate
from which the original claim follows by a routine summation argument. Observe from axiom (iv) and the triangle inequality that
for any .
Write for the logarithm function , thus for any . Without loss of generality we may assume that ; we then factor , where
This function is just when . When the function is more complicated, but we at least have the following crude bound:
Lemma 2 One has the pointwise bound .
Proof: We induct on . The case is obvious, so suppose and the claim has already been proven for . Since , we see from induction hypothesis and the triangle inequality that
Since by Möbius inversion, the claim follows.
We can write
In the region , we have . Thus
for . The contribution of the error term to to (10) is easily seen to be negligible if is large enough, so we may freely replace with with little difficulty.
If we insert this replacement directly into the left-hand side of (10) and rearrange, we get
We can’t quite control this using axiom (iv) because the range of is a bit too big, as explained in the introduction. So let us introduce a truncated function
where is a small quantity to be chosen later, and is a smooth function that equals on and equals on . Suppose one could establish the following two estimates for any fixed :
where is a quantity that depends on but not on . Then on combining the two estimates we would have
One could in principle compute explicitly from the proof of (13), but one can avoid doing so by the following comparison trick. In the special case , standard multiplicative number theory (noting that the Dirichlet series has a pole of order at , with top Laurent coefficient ) gives the asymptotic
which when compared with (14) for (recalling that in this case) gives the formula
Inserting this back into (14) and recalling that can be made arbitrarily small, we obtain (10).
As it turns out, the estimate (13) is easy to establish, but the estimate (12) is not, roughly speaking because the typical number in has too many divisors in the range , each of which gives a contribution to the error term. (In the book of Friedlander and Iwaniec, the estimate (13) is established anyway, but only after assuming a stronger version of (iv), roughly speaking in which is allowed to be as large as .) To resolve this issue, we will insert a preliminary sieve that will remove most of the potential divisors i the range (leaving only about such divisors on the average for typical ), making the analogue of (12) easier to prove (at the cost of making the analogue of (13) more difficult). Namely, if one can find a function for which one has the estimates
for some quantity that depends on but not on , then by repeating the previous arguments we will again be able to establish (10).
The key estimate is (16). As we shall see, when comparing with , the weight will cost us a factor of , but the term in the definitions of and will recover a factor of , which will give the desired bound since we are assuming .
One has some flexibility in how to select the weight : basically any standard sieve that uses divisors of size at most to localise (at least approximately) to numbers that are rough in the sense that they have no (or at least very few) factors less than , will do. We will use the analytic Selberg sieve choice
where is a smooth function supported on that equals on .
It remains to establish the bounds (15), (16), (17). To warm up and introduce the various methods needed, we begin with the standard bound
where denotes the derivative of . Note the loss of that had previously been pointed out. In the arguments that follows I will be a little brief with the details, as they are standard (see e.g. this previous post).
We now prove (19). The left-hand side can be expanded as
where denotes the least common multiple of and . From the support of we see that the summand is only non-vanishing when . We now use axiom (iv) and split the left-hand side into a main term
and an error term that is at most
From axiom (ii) and elementary multiplicative number theory, we have the bound
so from axiom (iv) and Cauchy-Schwarz we see that the error term (20) is acceptable. Thus it will suffice to establish the bound
The summand here is almost, but not quite, multiplicative in . To make it genuinely multiplicative, we perform a (shifted) Fourier expansion
for some rapidly decreasing function (essentially the Fourier transform of ). Thus
and so the left-hand side of (21) can be rearranged using Fubini’s theorem as
We can factorise as an Euler product:
Taking absolute values and using Mertens’ theorem leads to the crude bound
which when combined with the rapid decrease of , allows us to restrict the region of integration in (23) to the square (say) with negligible error. Next, we use the Euler product
for to factorise
For with nonnegative real part, one has
and so by the Weierstrass -test, is continuous at . Since
we thus have
Also, since has a pole of order at with residue , we have
The quantity (23) can thus be written, up to errors of , as
Using the rapid decrease of , we may remove the restriction on , and it will now suffice to prove the identity
But on differentiating and then squaring (22) we have
and the claim follows by integrating in from zero to infinity (noting that vanishes for ).
We have the following variant of (19):
Lemma 3 For any , one has
where the are such that
for any . We also have the variant
If in addition has no prime factors less than for some fixed , one has
Roughly speaking, the above estimates assert that is concentrated on those numbers with no prime factors much less than , but factors without such small prime divisors occur with about the same relative density as they do in the integers.
Proof: The left-hand side of (24) can be expanded as
If we define
then the previous expression can be written as
while one has
which gives (25) from Axiom (iv). To prove (24), it now suffices to show that
Arguing as before, the left-hand side is
From Mertens’ theorem we have
when , so the contribution of the terms where can be absorbed into the error (after increasing that error slightly). For the remaining contributions, we see that
where if does not divide , and
if divides times for some . In the latter case, Taylor expansion gives the bounds
and the claim (28) follows. When and we have
and (27) follows by repeating the previous calculations. Finally, (26) is proven similarly to (24) (using in place of ).
Now we can prove (15), (16), (17). We begin with (15). Using the Leibniz rule applied to the identity and using and Möbius inversion (and the associativity and commutativity of Dirichlet convolution) we see that
Next, by applying the Leibniz rule to for some and using (29) we see that
and hence we have the recursive identity
In particular, from induction we see that is supported on numbers with at most distinct prime factors, and hence is supported on numbers with at most distinct prime factors. In particular, from (18) we see that on the support of . Thus it will suffice to show that
If and , then has at most distinct prime factors , with . If we factor , where is the contribution of those with , and is the contribution of those with , then at least one of the following two statements hold:
- (a) (and hence ) is divisible by a square number of size at least .
- (b) .
The contribution of case (a) is easily seen to be acceptable by axiom (ii). For case (b), we observe from (30) and induction that
and so it will suffice to show that
where ranges over numbers bounded by with at most distinct prime factors, the smallest of which is at most , and consists of those numbers with no prime factor less than or equal to . Applying (26) (with replaced by ) gives the bound
so by (25) it suffices to show that
subject to the same constraints on as before. The contribution of those with distinct prime factors can be bounded by
applying Mertens’ theorem and summing over , one obtains the claim.
Now we show (16). As discussed previously in this section, we can replace by with negligible error. Comparing this with (16) and (11), we see that it suffices to show that
From the support of , the summand on the left-hand side is only non-zero when , which makes , where we use the crucial hypothesis to gain enough powers of to make the argument here work. Applying Lemma 2, we reduce to showing that
We can make the change of variables to flip the sum
and then swap the sums to reduce to showing that
By Lemma 3, it suffices to show that
To prove this, we use the Rankin trick, bounding the implied weight by . We can then bound the left-hand side by the Euler product
which can be bounded by
and the claim follows from Mertens’ theorem.
Finally, we show (17). By (11), the left-hand side expands as
We let be a small constant to be chosen later. We divide the outer sum into two ranges, depending on whether only has prime factors greater than or not. In the former case, we can apply (27) to write this contribution as
plus a negligible error, where the is implicitly restricted to numbers with all prime factors greater than . The main term is messy, but it is of the required form up to an acceptable error, so there is no need to compute it any further. It remains to consider those that have at least one prime factor less than . Here we use (24) instead of (27) as well as Lemma 3 to dominate this contribution by
up to negligible errors, where is now restricted to have at least one prime factor less than . This makes at least one of the factors to be at most . A routine application of Rankin’s trick shows that
and so the total contribution of this case is . Since can be made arbitrarily small, (17) follows.
— 2. Weierstrass approximation —
Having proved Theorem 1, we now take linear combinations of this theorem, combined with the Weierstrass approximation theorem, to give the asymptotics (7), (8) described in the introduction.
Let , , , be as in that theorem. It will be convenient to normalise the weights by to make their mean value comparable to . From Theorem 1 and summation by parts we have
whenever does not consist entirely of ones.
We now take a closer look at what happens when does consist entirely of ones. Let denote the -tuple . Convolving the case of (30) with copies of for some and using the Leibniz rule, we see that
Multiplying by and summing over , and using (31) to control the term, one has
If we define (up to an error of ) by the formula
then an induction then shows that
for odd , and
for even . In particular, after adjusting by if necessary, we have since the left-hand sides are non-negative.
If we now define the comparison sequence , standard multiplicative number theory shows that the above estimates also hold when is replaced by ; thus
for both odd and even . The bound (31) also holds for when does not consist entirely of ones, and hence
for any fixed (which may or may not consist entirely of ones).
Next, from induction (on ), the Leibniz rule, and (30), we see that for any and , , the function
is a finite linear combination of functions of the form for tuples that may possibly consist entirely of ones. We thus have
whenever is one of these functions (32). Specialising to the case , we thus have
where . The contribution of those that are powers of primes can be easily seen to be negligible, leading to
where now . The contribution of the case where two of the primes agree can also be seen to be negligible, as can the error when replacing with , and then by symmetry
By linearity, this implies that
for any polynomial that vanishes on the coordinate hyperplanes . The right-hand side can also be evaluated by Mertens’ theorem as
when is odd and
when is even. Using the Weierstrass approximation theorem, we then have
for any continuous function that is compactly supported in the interior of . Computing the right-hand side using Mertens’ theorem as before, we obtain the claimed asymptotics (7), (8).
Remark 4 The Bombieri asymptotic sieve has to use the full power of EH (or GEH); there are constructions due to Ford that show that if one only has a distributional hypothesis up to for some fixed constant , then the asymptotics of sums such as (5), or more generally (9), are not determined by a single scalar parameter , but can also vary in other ways as well. Thus the Bombieri asymptotic sieve really is asymptotic; in order to get type error terms one needs the level of distribution to be asymptotically equal to as . Related to this, the quantitative decay of the error terms in the Bombieri asymptotic sieve are extremely poor; in particular, they depend on the dependence of implied constant in axiom (iv) on the parameters , for which there is no consensus on what one should conjecturally expect.