Many problems in non-multiplicative prime number theory can be recast as sieving problems. Consider for instance the problem of counting the number of pairs of twin primes contained in for some large ; note that the claim that for arbitrarily large is equivalent to the twin prime conjecture. One can obtain this count by any of the following variants of the sieve of Eratosthenes:
- Let be the set of natural numbers in . For each prime , let be the union of the residue classes and . Then is the cardinality of the sifted set .
- Let be the set of primes in . For each prime , let be the residue class . Then is the cardinality of the sifted set .
- Let be the set of primes in . For each prime , let be the residue class . Then is the cardinality of the sifted set .
- Let be the set . For each prime , let be the residue class Then is the cardinality of the sifted set .
Exercise 1 Develop similar sifting formulations of the other three Landau problems.
In view of these sieving interpretations of number-theoretic problems, it becomes natural to try to estimate the size of sifted sets for various finite sets of integers, and subsets of integers indexed by primes dividing some squarefree natural number (which, in the above examples, would be the product of all primes up to ). As we see in the above examples, the sets in applications are typically the union of one or more residue classes modulo , but we will work at a more abstract level of generality here by treating as more or less arbitrary sets of integers, without caring too much about the arithmetic structure of such sets.
It turns out to be conceptually more natural to replace sets by functions, and to consider the more general the task of estimating sifted sums
for some finitely supported sequence of non-negative numbers; the previous combinatorial sifting problem then corresponds to the indicator function case . (One could also use other index sets here than the integers if desired; for much of sieve theory the index set and its subsets are treated as abstract sets, so the exact arithmetic structure of these sets is not of primary importance.)
Continuing with twin primes as a running example, we thus have the following sample sieving problem:
Problem 2 (Sieving problem for twin primes) Let , and let denote the number of natural numbers which avoid the residue classes for all primes . In other words, we have
where , is the product of all the primes strictly less than (we omit itself for minor technical reasons), and is the union of the residue classes . Obtain upper and lower bounds on which are as strong as possible in the asymptotic regime where goes to infinity and the sifting level grows with (ideally we would like to grow as fast as ).
From the preceding discussion we know that the number of twin prime pairs in is equal to , if is not a perfect square; one also easily sees that the number of twin prime pairs in is at least , again if is not a perfect square. Thus we see that a sufficiently good answer to Problem 2 would resolve the twin prime conjecture, particularly if we can get the sifting level to be as large as .
We return now to the general problem of estimating (1). We may expand
where (with the convention that ). We thus arrive at the Legendre sieve identity
Specialising to the case of an indicator function , we recover the inclusion-exclusion formula
Such exact sieving formulae are already satisfactory for controlling sifted sets or sifted sums when the amount of sieving is relatively small compared to the size of . For instance, let us return to the running example in Problem 2 for some . Observe that each in this example consists of residue classes modulo , where is defined to equal when and when is odd. By the Chinese remainder theorem, this implies that for each , consists of residue classes modulo . Using the basic bound
for any and any residue class , we conclude that
for any , where is the multiplicative function
Since and there are at most primes dividing , we may crudely bound , thus
Also, the number of divisors of is at most . From the Legendre sieve (3), we thus conclude that
We can factorise the main term to obtain
This is compatible with the heuristic
coming from the equidistribution of residues principle (Section 3 of Supplement 4), bearing in mind (from the modified Cramér model, see Section 1 of Supplement 4) that we expect this heuristic to become inaccurate when becomes very large. We can simplify the right-hand side of (7) by recalling the twin prime constant
(see equation (7) from Supplement 4); note that
so from Mertens’ third theorem (Theorem 42 from Notes 1) one has
as . Bounding crudely by , we conclude in particular that
when with . This is somewhat encouraging for the purposes of getting a sufficiently good answer to Problem 2 to resolve the twin prime conjecture, but note that is currently far too small: one needs to get as large as before one is counting twin primes, and currently can only get as large as .
The problem is that the number of terms in the Legendre sieve (3) basically grows exponentially in , and so the error terms in (4) accumulate to an unacceptable extent once is significantly larger than . An alternative way to phrase this problem is that the estimate (4) is only expected to be truly useful in the regime ; on the other hand, the moduli appearing in (3) can be as large as , which grows exponentially in by the prime number theorem.
To resolve this problem, it is thus natural to try to truncate the Legendre sieve, in such a way that one only uses information about the sums for a relatively small number of divisors of , such as those which are below a certain threshold . This leads to the following general sieving problem:
Problem 3 (General sieving problem) Let be a squarefree natural number, and let be a set of divisors of . For each prime dividing , let be a set of integers, and define for all (with the convention that ). Suppose that is an (unknown) finitely supported sequence of non-negative reals, whose sums
are known for all . What are the best upper and lower bounds one can conclude on the quantity (1)?
Here is a simple example of this type of problem (corresponding to the case , , , , and ):
Exercise 4 Let be a finitely supported sequence of non-negative reals such that , , and . Show that
and give counterexamples to show that these bounds cannot be improved in general, even when is an indicator function sequence.
Problem 3 is an example of a linear programming problem. By using linear programming duality (as encapsulated by results such as the Hahn-Banach theorem, the separating hyperplane theorem, or the Farkas lemma), we can rephrase the above problem in terms of upper and lower bound sieves:
Theorem 5 (Dual sieve problem) Let be as in Problem 3. We assume that Problem 3 is feasible, in the sense that there exists at least one finitely supported sequence of non-negative reals obeying the constraints in that problem. Define an (normalised) upper bound sieve to be a function of the form
for some coefficients , and obeying the pointwise lower bound
for all (in particular is non-negative). Similarly, define a (normalised) lower bound sieve to be a function of the form
for some coefficients , and obeying the pointwise upper bound
for all . Thus for instance and are (trivially) upper bound sieves and lower bound sieves respectively.
- (i) The supremal value of the quantity (1), subject to the constraints in Problem 3, is equal to the infimal value of the quantity , as ranges over all upper bound sieves.
- (ii) The infimal value of the quantity (1), subject to the constraints in Problem 3, is equal to the supremal value of the quantity , as ranges over all lower bound sieves.
Proof: We prove part (i) only, and leave part (ii) as an exercise. Let be the supremal value of the quantity (1) given the constraints in Problem 3, and let be the infimal value of . We need to show that .
We first establish the easy inequality . If the sequence obeys the constraints in Problem 3, and is an upper bound sieve, then
and hence (by the non-negativity of and )
taking suprema in and infima in we conclude that .
Now suppose for contradiction that , thus for some real number . We will argue using the hyperplane separation theorem; one can also proceed using one of the other duality results mentioned above. (See this previous blog post for some discussion of the connections between these various forms of linear duality.) Consider the affine functional
on the vector space of finitely supported sequences of reals. On the one hand, since , this functional is positive for every sequence obeying the constraints in Problem 3. Next, let be the space of affine functionals of the form
for some real numbers , some non-negative function which is a finite linear combination of the for , and some non-negative . This is a closed convex cone in a finite-dimensional vector space ; note also that lies in . Suppose first that , thus we have a representation of the form
for any finitely supported sequence . Comparing coefficients, we conclude that
for any (i.e., is an upper bound sieve), and also
and thus , a contradiction. Thus lies outside of . But then by the hyperplane separation theorem, we can find an affine functional on that is non-negative on and negative on . By duality, such an affine functional takes the form for some finitely supported sequence and (indeed, can be supported on a finite set consisting of a single representative for each atom of the finite -algebra generated by the ). Since is non-negative on the cone , we see (on testing against multiples of the functionals or ) that the and are non-negative, and that for all ; thus is feasible for Problem 3. Since is negative on , we see that
and thus , giving the desired contradiction.
Exercise 6 Prove part (ii) of the above theorem.
Exercise 7 Show that the infima and suprema in the above theorem are actually attained (so one can replace “infimal” and “supremal” by “minimal” and “maximal” if desired).
Exercise 8 What are the optimal upper and lower bound sieves for Exercise 4?
In the case when consists of all the divisors of , we see that the Legendre sieve is both the optimal upper bound sieve and the optimal lower bound sieve, regardless of what the quantities are. However, in most cases of interest, will only be some strict subset of the divisors of , and there will be a gap between the optimal upper and lower bounds.
Observe that a sequence of real numbers will form an upper bound sieve if one has the inequalities
for all ; we will refer to such sequences as upper bound sieve coefficients. (Conversely, if the sets are in “general position” in the sense that every set of the form for is non-empty, we see that every upper bound sieve arises from a sequence of upper bound sieve coefficients.) Similarly, a sequence of real numbers will form a lower bound sieve if one has the inequalities
for all with ; we will refer to such sequences as lower bound sieve coefficients.
Exercise 9 (Brun pure sieve) Let be a squarefree number, and a non-negative integer. Show that the sequence defined by
where is the number of prime factors of , is a sequence of upper bound sieve coefficients for even , and a sequence of lower bound sieve coefficients for odd . Deduce the Bonferroni inequalities
when is even, and
when is odd, whenever one is in the situation of Problem 3 (and contains all with ). The resulting upper and lower bound sieves are sometimes known as Brun pure sieves. The Legendre sieve can be viewed as the limiting case when .
In many applications the sums in (9) take the form
for some quantity independent of , some multiplicative function with , and some remainder term whose effect is expected to be negligible on average if is restricted to be small, e.g. less than a threshold ; note for instance that (5) is of this form if for some fixed (note from the divisor bound, Lemma 23 of Notes 1, that if ). We are thus led to the following idealisation of the sieving problem, in which the remainder terms are ignored:
Problem 10 (Idealised sieving) Let (we refer to as the sifting level and as the level of distribution), let be a multiplicative function with , and let . How small can one make the quantity
for a sequence of upper bound sieve coefficients, and how large can one make the quantity
for a sequence of lower bound sieve coefficients?
Thus, for instance, the trivial upper bound sieve and the trivial lower bound sieve show that (14) can equal and (15) can equal . Of course, one hopes to do better than these trivial bounds in many situations; usually one can improve the upper bound quite substantially, but improving the lower bound is significantly more difficult, particularly when is large compared with .
If the remainder terms in (13) are indeed negligible on average for , then one expects the upper and lower bounds in Problem 3 to essentially be the optimal bounds in (14) and (15) respectively, multiplied by the normalisation factor . Thus Problem 10 serves as a good model problem for Problem 3, in which all the arithmetic content of the original sieving problem has been abstracted into two parameters and a multiplicative function . In many applications, will be approximately on the average for some fixed , known as the sieve dimension; for instance, in the twin prime sieving problem discussed above, the sieve dimension is . The larger one makes the level of distribution compared to , the more choices one has for the upper and lower bound sieves; it is thus of interest to obtain equidistribution estimates such as (13) for as large as possible. When the sequence is of arithmetic origin (for instance, if it is the von Mangoldt function ), then estimates such as the Bombieri-Vinogradov theorem, Theorem 17 from Notes 3, turn out to be particularly useful in this regard; in other contexts, the required equidistribution estimates might come from other sources, such as homogeneous dynamics, or the theory of expander graphs (the latter arises in the recent theory of the affine sieve, discussed in this previous blog post). However, the sieve-theoretic tools developed in this post are not particularly sensitive to how a certain level of distribution is attained, and are generally content to use sieve axioms such as (13) as “black boxes”.
In some applications one needs to modify Problem 10 in various technical ways (e.g. in altering the product , the set , or the definition of an upper or lower sieve coefficient sequence), but to simplify the exposition we will focus on the above problem without such alterations.
As the exercise below (or the heuristic (7)) suggests, the “natural” size of (14) and (15) is given by the quantity (so that the natural size for Problem 3 is ):
Exercise 11 Let be as in Problem 10, and set .
- (i) Show that the quantity (14) is always at least when is a sequence of upper bound sieve coefficients. Similarly, show that the quantity (15) is always at most when is a sequence of lower bound sieve coefficients. (Hint: compute the expected value of when is a random factor of chosen according to a certain probability distribution depending on .)
- (ii) Show that (14) and (15) can both attain the value of when . (Hint: translate the Legendre sieve to this setting.)
The problem of finding good sequences of upper and lower bound sieve coefficients in order to solve problems such as Problem 10 is one of the core objectives of sieve theory, and has been intensively studied. This is more of an optimisation problem rather than a genuinely number theoretic problem; however, the optimisation problem is sufficiently complicated that it has not been solved exactly or even asymptotically, except in a few special cases. (It can be reduced to a optimisation problem involving multilinear integrals of certain unknown functions of several variables, but this problem is rather difficult to analyse further; see these lecture notes of Selberg for further discussion.) But while we do not yet have a definitive solution to this problem in general, we do have a number of good general-purpose upper and lower bound sieve coefficients that give fairly good values for (14), (15), often coming within a constant factor of the idealised value , and which work well for sifting levels as large as a small power of the level of distribution . Unfortunately, we also know of an important limitation to the sieve, known as the parity problem, that prevents one from taking as large as while still obtaining non-trivial lower bounds; as a consequence, sieve theory is not able, on its own, to sift out primes for such purposes as establishing the twin prime conjecture. However, it is still possible to use these sieves, in conjunction with additional tools, to produce various types of primes or prime patterns in some cases; examples of this include the theorem of Ben Green and myself in which an upper bound sieve is used to demonstrate the existence of primes in arbitrarily long arithmetic progressions, or the more recent theorem of Zhang in which (among other things) used an upper bound sieve was used to demonstrate the existence of infinitely many pairs of primes whose difference was bounded. In such arguments, the upper bound sieve was used not so much to count the primes or prime patterns directly, but to serve instead as a sort of “container” to efficiently envelop such prime patterns; when used in such a manner, the upper bound sieves are sometimes known as enveloping sieves. If the original sequence was supported on primes, then the enveloping sieve can be viewed as a “smoothed out indicator function” that is concentrated on almost primes, which in this context refers to numbers with no small prime factors.
In a somewhat different direction, it can be possible in some cases to break the parity barrier by assuming additional equidistribution axioms on the sequence than just (13), in particular controlling certain bilinear sums involving rather than just linear sums of the . This approach was in particular pursued by Friedlander and Iwaniec, leading to their theorem that there are infinitely many primes of the form .
The study of sieves is an immense topic; see for instance the recent 527-page text by Friedlander and Iwaniec. We will limit attention to two sieves which give good general-purpose results, if not necessarily the most optimal ones:
- (i) The beta sieve (or Rosser-Iwaniec sieve), which is a modification of the classical combinatorial sieve of Brun. (A collection of sieve coefficients is called combinatorial if its coefficients lie in .) The beta sieve is a family of upper and lower bound combinatorial sieves, and are particularly useful for efficiently sieving out all primes up to a parameter from a set of integers of size , in the regime where is moderately large, leading to what is sometimes known as the fundamental lemma of sieve theory.
- (ii) The Selberg upper bound sieve, which is a general-purpose sieve that can serve both as an upper bound sieve for classical sieving problems, as well as an enveloping sieve for sets such as the primes. (One can also convert the Selberg upper bound sieve into a lower bound sieve in a number of ways, but we will only touch upon this briefly.) A key advantage of the Selberg sieve is that, due to the “quadratic” nature of the sieve, the difficult optimisation problem in Problem 10 is replaced with a much more tractable quadratic optimisation problem, which can often be solved for exactly.
Remark 12 It is possible to compose two sieves together, for instance by using the observation that the product of two upper bound sieves is again an upper bound sieve, or that the product of an upper bound sieve and a lower bound sieve is a lower bound sieve. Such a composition of sieves is useful in some applications, for instance if one wants to apply the fundamental lemma as a “preliminary sieve” to sieve out small primes, but then use a more precise sieve like the Selberg sieve to sieve out medium primes. We will see an example of this in later notes, when we discuss the linear beta-sieve.
We will also briefly present the (arithmetic) large sieve, which gives a rather different approach to Problem 3 in the case that each consists of some number (typically a large number) of residue classes modulo , and is powered by the (analytic) large sieve inequality of the preceding section. As an application of these methods, we will utilise the Selberg upper bound sieve as an enveloping sieve to establish Zhang’s theorem on bounded gaps between primes. Finally, we give an informal discussion of the parity barrier which gives some heuristic limitations on what sieve theory is able to accomplish with regards to counting prime patters such as twin primes.
These notes are only an introduction to the vast topic of sieve theory; more detailed discussion can be found in the Friedlander-Iwaniec text, in these lecture notes of Selberg, and in many further texts.
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