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Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges“, submitted to Proceedings of the London Mathematical Society. This paper is concerned with the estimation of correlations such as

for medium-sized and large , where is the von Mangoldt function; we also consider variants of this sum in which one of the von Mangoldt functions is replaced with a (higher order) divisor function, but for sake of discussion let us focus just on the sum (1). Understanding this sum is very closely related to the problem of finding pairs of primes that differ by ; for instance, if one could establish a lower bound

then this would easily imply the twin prime conjecture.

The (first) Hardy-Littlewood conjecture asserts an asymptotic

as for any fixed positive , where the *singular series* is an arithmetic factor arising from the irregularity of distribution of at small moduli, defined explicitly by

when is even, and when is odd, where

is (half of) the twin prime constant. See for instance this previous blog post for a a heuristic explanation of this conjecture. From the previous discussion we see that (2) for would imply the twin prime conjecture. Sieve theoretic methods are only able to provide an upper bound of the form .

Needless to say, apart from the trivial case of odd , there are no values of for which the Hardy-Littlewood conjecture is known. However there are some results that say that this conjecture holds “on the average”: in particular, if is a quantity depending on that is somewhat large, there are results that show that (2) holds for most (i.e. for ) of the betwen and . Ideally one would like to get as small as possible, in particular one can view the full Hardy-Littlewood conjecture as the endpoint case when is bounded.

The first results in this direction were by van der Corput and by Lavrik, who established such a result with (with a subsequent refinement by Balog); Wolke lowered to , and Mikawa lowered further to . The main result of this paper is a further lowering of to . In fact (as in the preceding works) we get a better error term than , namely an error of the shape for any .

Our arguments initially proceed along standard lines. One can use the Hardy-Littlewood circle method to express the correlation in (2) as an integral involving exponential sums . The contribution of “major arc” is known by a standard computation to recover the main term plus acceptable errors, so it is a matter of controlling the “minor arcs”. After averaging in and using the Plancherel identity, one is basically faced with establishing a bound of the form

for any “minor arc” . If is somewhat close to a low height rational (specifically, if it is within of such a rational with ), then this type of estimate is roughly of comparable strength (by another application of Plancherel) to the best available prime number theorem in short intervals on the average, namely that the prime number theorem holds for most intervals of the form , and we can handle this case using standard mean value theorems for Dirichlet series. So we can restrict attention to the “strongly minor arc” case where is far from such rationals.

The next step (following some ideas we found in a paper of Zhan) is to rewrite this estimate not in terms of the exponential sums , but rather in terms of the Dirichlet polynomial . After a certain amount of computation (including some oscillatory integral estimates arising from stationary phase), one is eventually reduced to the task of establishing an estimate of the form

for any (with sufficiently large depending on ).

The next step, which is again standard, is the use of the Heath-Brown identity (as discussed for instance in this previous blog post) to split up into a number of components that have a Dirichlet convolution structure. Because the exponent we are shooting for is less than , we end up with five types of components that arise, which we call “Type “, “Type “, “Type “, “Type “, and “Type II”. The “Type II” sums are Dirichlet convolutions involving a factor supported on a range and is quite easy to deal with; the “Type ” terms are Dirichlet convolutions that resemble (non-degenerate portions of) the divisor function, formed from convolving together portions of . The “Type ” and “Type ” terms can be estimated satisfactorily by standard moment estimates for Dirichlet polynomials; this already recovers the result of Mikawa (and our argument is in fact slightly more elementary in that no Kloosterman sum estimates are required). It is the treatment of the “Type ” and “Type ” sums that require some new analysis, with the Type terms turning to be the most delicate. After using an existing moment estimate of Jutila for Dirichlet L-functions, matters reduce to obtaining a family of estimates, a typical one of which (relating to the more difficult Type sums) is of the form

for “typical” ordinates of size , where is the Dirichlet polynomial (a fragment of the Riemann zeta function). The precise definition of “typical” is a little technical (because of the complicated nature of Jutila’s estimate) and will not be detailed here. Such a claim would follow easily from the Lindelof hypothesis (which would imply that ) but of course we would like to have an unconditional result.

At this point, having exhausted all the Dirichlet polynomial estimates that are usefully available, we return to “physical space”. Using some further Fourier-analytic and oscillatory integral computations, we can estimate the left-hand side of (3) by an expression that is roughly of the shape

The phase can be Taylor expanded as the sum of and a lower order term , plus negligible errors. If we could discard the lower order term then we would get quite a good bound using the exponential sum estimates of Robert and Sargos, which control averages of exponential sums with purely monomial phases, with the averaging allowing us to exploit the hypothesis that is “typical”. Figuring out how to get rid of this lower order term caused some inefficiency in our arguments; the best we could do (after much experimentation) was to use Fourier analysis to shorten the sums, estimate a one-parameter average exponential sum with a binomial phase by a two-parameter average with a monomial phase, and then use the van der Corput process followed by the estimates of Robert and Sargos. This rather complicated procedure works up to it may be possible that some alternate way to proceed here could improve the exponent somewhat.

In a sequel to this paper, we will use a somewhat different method to reduce to a much smaller value of , but only if we replace the correlations by either or , and also we now only save a in the error term rather than .

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

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:

- (i) (Non-negativity) One has for all .
- (ii) (Crude size bound) One has for all , where is the divisor function.
- (iii) (Size) We have for some constant .
- (iv) (Elliott-Halberstam type conjecture) For any , one has
where is a multiplicative function with for all primes and .

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 1Let be as above. Let be a tuple of natural numbers (independent of ) that is not equal to . Then one has the asymptoticas , 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:

*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

where

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

and thus

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):

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

where

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

and hence

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 4The 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.

The twin prime conjecture is one of the oldest unsolved problems in analytic number theory. There are several reasons why this conjecture remains out of reach of current techniques, but the most important obstacle is the parity problem which prevents purely sieve-theoretic methods (or many other popular methods in analytic number theory, such as the circle method) from detecting pairs of prime twins in a way that can distinguish them from other twins of almost primes. The parity problem is discussed in these previous blog posts; this obstruction is ultimately powered by the *Möbius pseudorandomness principle* that asserts that the Möbius function is asymptotically orthogonal to all “structured” functions (and in particular, to the weight functions constructed from sieve theory methods).

However, there is an intriguing “alternate universe” in which the Möbius function *is* strongly correlated with some structured functions, and specifically with some Dirichlet characters, leading to the existence of the infamous “Siegel zero“. In this scenario, the parity problem obstruction disappears, and it becomes possible, *in principle*, to attack problems such as the twin prime conjecture. In particular, we have the following result of Heath-Brown:

Theorem 1At least one of the following two statements are true:

- (Twin prime conjecture) There are infinitely many primes such that is also prime.
- (No Siegel zeroes) There exists a constant such that for every real Dirichlet character of conductor , the associated Dirichlet -function has no zeroes in the interval .

Informally, this result asserts that if one had an infinite sequence of Siegel zeroes, one could use this to generate infinitely many twin primes. See this survey of Friedlander and Iwaniec for more on this “illusory” or “ghostly” parallel universe in analytic number theory that should not actually exist, but is surprisingly self-consistent and to date proven to be impossible to banish from the realm of possibility.

The strategy of Heath-Brown’s proof is fairly straightforward to describe. The usual starting point is to try to lower bound

for some large value of , where is the von Mangoldt function. Actually, in this post we will work with the slight variant

where

is the second von Mangoldt function, and denotes Dirichlet convolution, and is an (unsquared) Selberg sieve that damps out small prime factors. This sum also detects twin primes, but will lead to slightly simpler computations. For technical reasons we will also smooth out the interval and remove very small primes from , but we will skip over these steps for the purpose of this informal discussion. (In Heath-Brown’s original paper, the Selberg sieve is essentially replaced by the more combinatorial restriction for some large , where is the primorial of , but I found the computations to be slightly easier if one works with a Selberg sieve, particularly if the sieve is not squared to make it nonnegative.)

If there is a Siegel zero with close to and a Dirichlet character of conductor , then multiplicative number theory methods can be used to show that the Möbius function “pretends” to be like the character in the sense that for “most” primes near (e.g. in the range for some small and large ). Traditionally, one uses complex-analytic methods to demonstrate this, but one can also use elementary multiplicative number theory methods to establish these results (qualitatively at least), as will be shown below the fold.

The fact that pretends to be like can be used to construct a tractable approximation (after inserting the sieve weight ) in the range (where for some large ) for the second von Mangoldt function , namely the function

Roughly speaking, we think of the periodic function and the slowly varying function as being of about the same “complexity” as the constant function , so that is roughly of the same “complexity” as the divisor function

which is considerably simpler to obtain asymptotics for than the von Mangoldt function as the Möbius function is no longer present. (For instance, note from the Dirichlet hyperbola method that one can estimate to accuracy with little difficulty, whereas to obtain a comparable level of accuracy for or is essentially the Riemann hypothesis.)

One expects to be a good approximant to if is of size and has no prime factors less than for some large constant . The Selberg sieve will be mostly supported on numbers with no prime factor less than . As such, one can hope to approximate (1) by the expression

as it turns out, the error between this expression and (1) is easily controlled by sieve-theoretic techniques. Let us ignore the Selberg sieve for now and focus on the slightly simpler sum

As discussed above, this sum should be thought of as a slightly more complicated version of the sum

Accordingly, let us look (somewhat informally) at the task of estimating the model sum (3). One can think of this problem as basically that of counting solutions to the equation with in various ranges; this is clearly related to understanding the equidistribution of the hyperbola in . Taking Fourier transforms, the latter problem is closely related to estimation of the Kloosterman sums

where denotes the inverse of in . One can then use the Weil bound

where is the greatest common divisor of (with the convention that this is equal to if vanish), and the decays to zero as . The Weil bound yields good enough control on error terms to estimate (3), and as it turns out the same method also works to estimate (2) (provided that with large enough).

Actually one does not need the full strength of the Weil bound here; any power savings over the trivial bound of will do. In particular, it will suffice to use the weaker, but easier to prove, bounds of Kloosterman:

Lemma 2 (Kloosterman bound)One has

whenever and are coprime to , where the is with respect to the limit (and is uniform in ).

*Proof:* Observe from change of variables that the Kloosterman sum is unchanged if one replaces with for . For fixed , the number of such pairs is at least , thanks to the divisor bound. Thus it will suffice to establish the fourth moment bound

The left-hand side can be rearranged as

which by Fourier summation is equal to

Observe from the quadratic formula and the divisor bound that each pair has at most solutions to the system of equations . Hence the number of quadruples of the desired form is , and the claim follows.

We will also need another easy case of the Weil bound to handle some other portions of (2):

Lemma 3 (Easy Weil bound)Let be a primitive real Dirichlet character of conductor , and let . Then

*Proof:* As is the conductor of a primitive real Dirichlet character, is equal to times a squarefree odd number for some . By the Chinese remainder theorem, it thus suffices to establish the claim when is an odd prime. We may assume that is not divisible by this prime , as the claim is trivial otherwise. If vanishes then does not vanish, and the claim follows from the mean zero nature of ; similarly if vanishes. Hence we may assume that do not vanish, and then we can normalise them to equal . By completing the square it now suffices to show that

whenever . As is on the quadratic residues and on the non-residues, it now suffices to show that

But by making the change of variables , the left-hand side becomes , and the claim follows.

While the basic strategy of Heath-Brown’s argument is relatively straightforward, implementing it requires a large amount of computation to control both main terms and error terms. I experimented for a while with rearranging the argument to try to reduce the amount of computation; I did not fully succeed in arriving at a satisfactorily minimal amount of superfluous calculation, but I was able to at least reduce this amount a bit, mostly by replacing a combinatorial sieve with a Selberg-type sieve (which was not needed to be positive, so I dispensed with the squaring aspect of the Selberg sieve to simplify the calculations a little further; also for minor reasons it was convenient to retain a tiny portion of the combinatorial sieve to eliminate extremely small primes). Also some modest reductions in complexity can be obtained by using the second von Mangoldt function in place of . These exercises were primarily for my own benefit, but I am placing them here in case they are of interest to some other readers.

We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the *linear sieve* in which the sieve dimension is equal to , which is one of the best understood sieving situations, and one of the rare cases in which the precise limits of the sieve method are known. A bit more specifically, let be quantities with for some fixed , and let be a multiplicative function with

for all primes and some fixed (we allow all constants below to depend on ). Let , and for each prime , let be a set of integers, with for . We consider finitely supported sequences of non-negative reals for which we have bounds of the form

for all square-free and some , and some remainder terms . One is then interested in upper and lower bounds on the quantity

The fundamental lemma of sieve theory (Corollary 19 of Notes 4) gives us the bound

This bound is strong when is large, but is not as useful for smaller values of . We now give a sharp bound in this regime. We introduce the functions by

where we adopt the convention . Note that for each one has only finitely many non-zero summands in (6), (7). These functions are closely related to the Buchstab function from Exercise 28 of Supplement 4; indeed from comparing the definitions one has

for all .

Exercise 1 (Alternate definition of )Show that is continuously differentiable except at , and is continuously differentiable except at where it is continuous, obeying the delay-differential equationsfor , with the initial conditions

for and

for . Show that these properties of determine completely.

For future reference, we record the following explicit values of :

We will show

Theorem 2 (Linear sieve)Let the notation and hypotheses be as above, with . Then, for any , one has the upper boundif is sufficiently large depending on . Furthermore, this claim is sharp in the sense that the quantity cannot be replaced by any smaller quantity, and similarly cannot be replaced by any larger quantity.

Comparing the linear sieve with the fundamental lemma (and also testing using the sequence for some extremely large ), we conclude that we necessarily have the asymptotics

for all ; this can also be proven directly from the definitions of , or from Exercise 1, but is somewhat challenging to do so; see e.g. Chapter 11 of Friedlander-Iwaniec for details.

Exercise 3Establish the integral identitiesand

for . Argue heuristically that these identities are consistent with the bounds in Theorem 2 and the Buchstab identity (Equation (16) from Notes 4).

Exercise 4Use the Selberg sieve (Theorem 30 from Notes 4) to obtain a slightly weaker version of (12) in the range in which the error term is worsened to , but the main term is unchanged.

We will prove Theorem 2 below the fold. The optimality of is closely related to the parity problem obstruction discussed in Section 5 of Notes 4; a naive application of the parity arguments there only give the weak bounds and for , but this can be sharpened by a more careful counting of various sums involving the Liouville function .

As an application of the linear sieve (specialised to the ranges in (10), (11)), we will establish a famous theorem of Chen, giving (in some sense) the closest approach to the twin prime conjecture that one can hope to achieve by sieve-theoretic methods:

Theorem 5 (Chen’s theorem)There are infinitely many primes such that is the product of at most two primes.

The same argument gives the version of Chen’s theorem for the even Goldbach conjecture, namely that for all sufficiently large even , there exists a prime between and such that is the product of at most two primes.

The discussion in these notes loosely follows that of Friedlander-Iwaniec (who study sieving problems in more general dimension than ).

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