1. Basic Sieve Methods and Applications

SIEVE METHODS LECTURE NOTES, SPRING 2020

KEVIN FORD

1. BASICSIEVEMETHODSANDAPPLICATIONS A sieve is a technique for bounding the size of a set after the elements with “undesirable properties” (usually of a number theoretic nature) have been removed. The undesirable properties could be divisibility by a prime from a given set, other multiplicative constraints (divisibility by a perfect square for example) or inclusion in a set of residue classes. Inclusion-exclusion yields an exact formula, however fork properties this produces2 k summands which is usually too much to effectively deal with. A sieve is a procedure to O 1 estimate the number of “desirable” elements of the set usingk p q summands. While inexact, oftentimes the sieve is capable of estimating the size very accurately. The original sieve is, of course, the Sieve of Eratosthenes, the familiar process of creating a table of prime numbers by systematically removing those integers divisible by small primes (but keeping the primes them- selves). The modern sieve was created by Viggo Brun in the period 1915-1922 as a way of attacking famous unsolved problems such as Golbach’s Conjecture and the Twin Prime problem (both, so far, unsuccessfully). Sieve methods have since found enormous application in number theory, often used as tools in many other types of problems, e.g. in studying Diophantine equations.

1.1. Notational conventions.τ n is the number of positive divisors ofn p q ω n is the number of distinct prime factors ofn p q Ω n is the number of prime factors ofn counted with multiplicity p q µ n is the Mobius’s¨ function;µ n 1 ω n ifn is squarefree andµ n 0 otherwise. p q p q “ p´ q p q p q “ P n is the largest prime factor ofn;P 1 0 by convention `p q `p q “ P n is the smallest prime factor ofn;P 1 by convention ´p q ´p q “ 8 P z is the set of positive squarefree integers composed only of primes z p q ď Λ n denotes the von Mangoldt function p q � n is the indicator function of the statementX or of the setX X p q the symbolp, with or without subscripts, always denotes a prime π x;q,a is the number of primesp x in the progressiona modq. p q ď P denotes probability andE expectation general 1.2. General sieve setup.A sieve problem is a probability space Ω, ,P , together with a “total mass” A p F q quantityM and events , one for each primep. The “sifting function” is A p S , z M P not p, p z . pA q“ ¨ p 1 A @ ď q 2 KEVIN FORD

OftentimesΩ will be afinite set of integers, withP n Ω 1 Ω for eachn Ω (the uniform probability p P q “ {| | P measure),M Ω and p is the event thatp n, that is, p n Ω:p n . This is the standard “small “| | A | A “ t P | u sieve” problem, whereS , z is the number ofn Ω withp�n for allp z; these are called the “unsifted pA q P ď numbers”. In the above definitions,M can be anything, but in practice it has some arithmetical meaning. Some specific examples: Eratosthenes sieve for primes.Ω 1, x N, uniform probabilities onΩ,M Ω txu, and ‚ “r s X “| | “ p n Ω:p n . ThenS , ?x π x π ?x 1, as the unsifted elements are the primes A “ t P | u pA q “ p q ´ p q ` in ?x, x together with the number 1. p s Twin primes.Ω 1, x N, p n Ω:p n n 2 . The unsifted numbers are numbersn ‚ “r s X A “ t P | p ` qu such thatn n 2 has no prime factor z. In particular,S , ?x 2 countsk ?x 2, x p ` q ď pA ` q Pp ` s for which bothk andk 2 are prime. ` Equivalently, p is the set ofn that avoid the residue classes 0 modp and 2 modp. A ´ Twin primes, weighted version.Ω 2, x N, and ‚ “r s X Λ n 2 P n Ω p ` q ,M Λ n 2 , M p P q “ “ 2 n x p ` q ďÿď p n Ω:p n . HereM 2 n x Λ n 2 x by the Prime Number Theorem, and A “ t P | u “ ď ď p ` q „ S , ?x ř Λ n 2 Λ p 2 pA q “ p ` q “ p ` q 2 n x ?x p x P ďÿn ď?x ÿă ď ´p qą 1 2 log p 2 O x { , “ p ` q ` p q ?x p x p 2ÿă primeď ` the error term coming from termsp 2 q b whereq is prime andb 2. ` “ ě 1 Prime tuples. Leta 1, ,a k N andb 1, . . . , bk Z. PutΩ 1, x N, and ‚ ¨¨¨ P P “r s X

p n Ω:p a 1n b 1 a n b . A “ t P |p ` q ¨ ¨ ¨ p k ` kqu

For an appropriatec 0, which depends ona 1, a2, . . . , a , b ,S , c ?x counts thosen x for ą k k pA q ď whicha 1n b 1, . . . , a n b are simultaneously prime. ` k ` k Prime values of a polynomial. Letf:Z Z be an irreducible polynomial of degreeh 1, put ‚ Ñ hě Ω 1, x Z, p n Ω:p f n . LetC be large enough so that h n Cn for alln 1. “r s X A “ t P | p qu | p q|ď ě Then, as before,S , ?Cxh captures values ofn for whichf n is prime. pA q p q Goldbach’s problem. LetN be an even, positive integer, putΩ 1,2,...,N 1 ,M Ω ‚ “t ´ u “| | “ N 1, and p k Ω:p k N k . ThenS , ?N counts numbersk ?N,N for which ´ A “ t P | p ´ qu pA q Pp s bothk andN k are prime. In particular,S , ?N 0 implies thatN is the sum of two primes. ´ pA qą If one shows this for allN 4, one deduces Goldbach’s Conjecture. ě

1Unless otherwise speciﬁed, from now on wheneverΩ is aﬁnite set, the probability measure onΩ will be the uniform measure, andM will be the number of elements ofΩ. SIEVE METHODS LECTURE NOTES, SPRING 2020 3

Primes in an arithmetic progression. Fix coprime positive integersa andq, let ‚ Ω 1 n x:n a modq , “t ď ď ” p qu

p n Ω:p n . ThenS , ?x captures primes in ?x, x that are in the arithmetic progres- A “ t P | u pA q p s siona modq. Sums of two squares.Ω 1, x Z, ‚ “r s X n Ω:p e n for some odde p 3 mod 4 p t P } u ” p q A “ otherwise. #H ThenS , x counts integersn x, for which we don’t havep e n for any primep 3 mod 4 pA q ď } ” p q and odd exponente. That is,S , x is the number of integersn x which are the sum of two pA q ď squares. Sieve by multiple residue classes. LetM,N be two integers,Ω N 1,M 2,...,N M ‚ “t ` ` ` u and for each primep let be some subset (possibly empty) of the residue classes modulop. Put I p

p n Ω:n modp p . A “ t P RI u

HereS , z counts the integers in N,M N avoiding all the residue classes p for primes pA q p ` s I p z. If p is bounded or bounded on average, then this is a very general sieving problem of ď |I | “small sieve” type, whereas if p is unbounded on average, the problem falls under the umbrella |I | of the “large sieve”. The case of prime values of a polynomial, see above, is a special case with

p n Z pZ:f n 0 modp . I “ t P { p q ” p qu Multivariate polynomial sieve. LetF x :Z k Z k be a multivariate polynomial ofx ‚ k p q Ñ “ x1, . . . , xk , take anyfiniteΩ Z , and p x Ω:p F x . ThenS , z countsx Ω for p q P A “ t P | p qu pA q P whichF x has no prime factorp z. p q ď The square-free sieve. LetΩ be afinite set of integers and for each primep let p n ‚ A “ t P Ω:p 2 n . ThenS , z , with an appropriately largez, will count the elements ofΩ which are | u pA q squarefree. A famous application is for squarefree values of a polynomial, e.g.Ω f n :1 “t p q ď n x , wheref is an irreducible polynomial. ď u One can similarly set up ak-free sieve problem. Elliptic curve sieve. Fix and elliptic curveE overQ LetΩ be the set of primesq x. Let ‚ ď p q Ω:p #E F q , whereE F q is the reduction ofE moduloq. It is known that#E F q A “ t P | { u { { ď q 1 2 ?q, and thusS ,2 ?x counts thoseq for which#E F q is prime. ` ` pA q { Ifd is a square-free integer composed only of primes in , define P Ad (1.1) d p,Ad M P d . A “ A “ ¨ pA q p d č| In particular,A 1 M. In the case whereΩ is afinite set of integers with uniform measure,M Ω , and “ “| | p n Ω:p n ,A counts the number ofn Ω divisible byd. A “ t P | u d P 4 KEVIN FORD

In this notation, inclusion-exclusion gives IE (1.2) S , z µ d A . pA q“ p q d P d z `ÿp qď 1.3. Legendre’s sieve for primes (Legendre, 1808). LetΩ 1, x Z, p n Ω:p n . Then “r s X A “ t P | u S , z counts the positive integersn x with no prime factor z; in particular, this includes all of the pA q ď ď primes betweenz andx. Also, for each aquarefreed, n Ω:d n and thus A d “ t P | u Ad d tx du x d O 1 . “ |A | “ { “ { ` p q By (1.2), x π x π z S , z µ d O 1 p q ´ p qď pA q“ p q d ` p q P d z `ÿp qď ´ ¯ µ d x p q O τ P “ d ` p p qq P d z `ÿp qď 1 π z x 1 O 2 p q . “ p z ´ p ` źď ˆ ˙ ´ ¯ Takingz logx, and using the crude boundπ z z together with Mertens’ bound, we conclude that “ p qď x γ logx x π x z e´ o 1 O 2 O . p qď ` logz ` p q ` p q “ log logx ˆ ˙ Remark: We can do better by observing` that in factA˘ 0 ford x (cf. Hooley [25]), and thus restricting d “ ą the sums to suchd.

1.4. The role of independence. If the events p are independent, then the sieving problem is trivial, for A then prob-approx (1.3) S , z M 1 P p . pA q“ p zp ´ A q źď In practice, the events are not independent, but are close to being so, especially for small primes. For exam-

ple, in Legendre’s sieve for primes,A d tx du is very close tox d. Even with this strong approximation, “ { { however, the accumulation of error terms (coming fromk-correlations) becomes unwieldy whenz is much larger than logx.

1.5. Main goals. We will see that the right side of (1.3) is a good approximation toS , z under very pA q general conditions. We will accomplish this by a judicious pruning of the summands in (1.2). To set things up, we adopt some additional notation. TakeX to be an approximation ofM (this can be anything, but in practice it is very close toM). We assume that there is a functiong satisfying g (g) 0 g p 1 all primesp , ď p qă p q which, when extended to a multiplicative function byg d p d g p , givesA d Xg d for squarefree p q “ | p q « p q d ; that is, we require that the “remainders” ś r (r) r : A Xg d d “ d ´ p q SIEVE METHODS LECTURE NOTES, SPRING 2020 5

to be “small on average”. In the case where the events p are independent, takingg p P p yieldsr d 0 A p q “ A “ for alld and we recover (1.3). In practice, however, we will not takeg p P p but something very close p q “ A which is convenient for calculations. We also adopt the short-hand

V (V) V z 1 g p . p q“ p zp ´ p qq źď In this notation, the right side of (1.3) is aboutXV z . p q Broadly speaking, ifr is not more than1 X on average over “small”d, we will be able to prove the d { following: o 1 (asymptotic)S , z XV z z X p q ; pA q„ p q p “ q sieve_goals (1.4) (upper bound)S , z XV z z X ; pA q! p q p ď q (lower bound)S , z XV z z X c , pA q" p q p ď q where the constantc depends on the nature of the sieve problem . A In plain language, we can prove the expected asymptotic formula for smallz, an upper bound of the expected order for allz, and a lower bound of the expected order ifz is at most a small power ofX. The upper bound in (1.4) is amazing in its generality, and it has enormous utility as an auxilliary counting device in many problems.

1.6. Sifting density / dimension, and level of distribution. For most sieving problems, we haveg p p q « κ p on average overp (we’ll make this precise below). In this caseκ is referred to as the sifting density { or sifting dimension. In the literature, the linear sieve refers to dimension 1. Various sieving procedures have been optimized for sieves of a particular density, e.g. the Rosser-Iwaniec theory of the linear sieve, and Iwaniec’s theory of the half-dimensional sieve. Roughly speaking, the level of distribution of a sieve problem is the smallestD for which A µ2 d r εXV z , p q| d|ď p q d D,P d z ď ÿ`p qď for an appropriate, smallε 0. The larger the level of distribution, the better quality of the bounds we can ą prove in (1.4).

1.7. More examples. There are limitations of the sieve, which are illustrated by the following examples.

1.7.1. Eratosthenes sieve. . The asymptotic in (1.4) cannot be expected to hold forz being aﬁxed power ofX. TakeΩ 1, x Z,X x, and set p n Ω:p n . As before, “r s X “ A “ t P | u Ad # n x:d n tx du x d O 1 “ t ď | u “ { “ { ` p q Thus, takingg d 1 d, we see that the error termsr O 1 . Also, p q “ { d “ p q x S , ?x π x π ?x 1 pA q “ p q ´ p q ` „ logx 6 KEVIN FORD

by the Prime Number Theorem. However, Mertens’ theorem gives 1 2e γx XV z x 1 ´ , p q“ ´ p „ logx p ?x ˆ ˙ ďź γ with2e ´ 1.122.... The discrepancy betweenS , ?x andXV z comes from the large amount of “ pA q p q 1 3 dependence among the events p for large primesp; e.g. p ifp p p x . In A A XA p1 XA p2 “H ą 1 ą 2 ą { fact, forﬁxedc 0, one hasS , x c w c XV z , wherew c 1 for almost allc 0,1 . Thus, in ą pA q „ p q p q p q ‰ Pp s general the conditionz X o 1 is necessary in order to conclude the asymptotic in(1.4). “ p q

1.7.2. Twin primes. . TakeΩ 1, x Z,X x, p n Ω:p n n 2 . HereS , ?x 2 “r s X “ A “ t P | p ` qu pA ` q counts the number of twin prime pairs between ?x 2 andx. Hardy and Littlewood [22, Conjecture B] ` conjectured that the count of such pairs is Cx log 2 x where „ { 1 C 2 1 1.32. “ ´ p 1 2 « p 2 ˆ ˙ źą p ´ q For a heuristic explanation of this formula, see Section 1.10 below. By breaking up 1, x into subintervals r s of lengthd, we easily derive ρ d A # n x:d n n 2 x p q O ρ d , d “ t ď | p ` qu “ d ` p p qq where ρ d # 0 k d 1:k k 2 0 modd . p q “ t ď ď ´ p ` q ” p qu By the Chinese remainder theorem,ρ is multiplicative,ρ 2 1 andρ p 2 forp 2. Thus, this is a p q “ p q “ ą sieve problem of dimension 2 (or sifting density 2). Puttingg d ρ d d and applying Mertens, we get p q “ p q{ that 2γ x 2 4e´ Cx twin_prime_Vz (1.5) XV z 1 p q . p q“ 2 ´ p „ log2 x 3 p z ˆ ˙ ďźď As in Example (a) above,XV z differs by a constant multiplicative factor from what is expected to be true. p q Sieve methods deliver an upper bound of the expected order (takez ?x) “ x # n x:n andn 2 are both prime xV z . t ď ` u ! p q— log2 x Sieve methods only, however, deliver a lower bound for somewhat smallerz. For example, it is known that 1 5 x S , x { XV z . pA q " p q— log2 x From the last estimate, we conclude that there are x log 2 x values ofk x for which each ofk andk 2 " { ď ` has at most 4 prime factors. We will prove a somewhat weaker version of this below (with 4 replaced by 10). This is a typical conclusion from a lower bound sieve. This is proved using the “linear variant” of the

twin prime sieving problem. Here takeΩ p 2:p x a set of shifted primes, p n Ω:p n . “t ` ď u A “ t P | u In this set-up, li x Ad π x; d, 2 p q 2�d “ p ´ q „ φ d p q p q SIEVE METHODS LECTURE NOTES, SPRING 2020 7

Admissible Non-admissible n, n 2k ;k N n, n 1 p ` q P p ` q n,2n 1 n,3n 1 p ` q p ` q n, n 2, n 6 n, n 2, n 4 p ` ` q p ` ` q by the prime number theorem in arithmetic progressions. For the application to the sieve, we need a uniform 1 estimate onπ x; d, 2 , at least on average overd. Here we takeX li x , adg p p 1 for oddp. p ´ q “ p q p q “ ´ Sinceg p 1 p, this is a sieve problem of dimension 1. Oftentimes, this process of “dimension lowering” p q « { yields stronger results, as the sieve procedures for smaller dimension are at present stronger.

1.7.3. Primek-tuples. . Leta 1, ,a k N andb 1, . . . , bk Z. PutΩ 1, x N, and ¨¨¨ P P “r s X p n Ω:p a 1n b 1 a n b . A “ t P |p ` q ¨ ¨ ¨ p k ` kqu An easy counting yields ρ d A x p q O ρ d , d “ d ` p p qq where

ρ d # 0 n d: a 1n b 1 a n b 0 modd . p q “ t ď ă p ` q ¨ ¨ ¨ p k ` kq ” p qu By the Chinese Remainder Theorem,ρ d is multiplicative. We say that the collection of linear forms p q a1n b 1, . . . , a n b is admissible ifρ p p for all primesp. In the contrary case, for alln, one of ` k ` k p qă the formsa jn b j is divisible byp and hence there are onlyﬁnitely manyn making all of thea jn b j ` ` simultaneously prime. Some examples: We setg p ρ p p. Note that if p q “ p q{ k p� ai aibj a jbi , p ´ q i 1 1 i j k ź“ ďźă ď thenρ p k (see Exercise 2.3 below). In particular,ρ p k for all sufﬁciently largep, and hence this is p q “ p q “ a sieve problem of densityk (or dimensionk). At present, sieve methods can deliver bounds of the form x # n x:a jn b j are prime for allj C , t ď ` uď logx k p q whereC is an explicit function ofa 1, b1, . . . , a , b . Ifk 2 we still do not know how to show that the left k k ě side goes to asx . 8 Ñ8

1.7.4. Selberg’s examples.[33]. The values ofc for which sieve method deliver lower bounds (1.4) are invariable smaller than we would like. There is a fundamental barrier at work which explains this, known as the “parity barrier”. Roughly speaking, the small sieve works with inputsX, the functiong and esti- 1 o 1 mates for the remaindersr d. However, even if the level of distribution is very large, sayx ´ p q, the sieve fundamentally cannot distinguish between numbers with an odd number of prime factors from those with 8 KEVIN FORD

an even number of prime factors. Consider two sequences deﬁned as follows. Recall the Liouville func- tionλ n 1 Ω n (the completely multiplicative function which is -1 at all primes). Let , be p q “ p´ q p q A ` A ´ probability spaces on the sets

Ω` 1 n x:λ n 1 ,Ω ´ 1 n x:λ n 1 , “ t ď ď p q “ u “ t ď ď p q “ ´ u

respectively, and p n Ω:p n for eachp. A “ t P | u The prime number theorem (with classical error term) implies

c ?logx λ n O xe ´ 1 , somec 1 0. n x p q “ p q ą ÿď Therefore, 1 λ dm 1 x λ d # n Ω ˘ :d n ˘ p q p q λ m t P | u “ 2 “ 2 d ˘ 2 p q m x d m x d ÿď { Y ] ÿď { x x c?log x d O e´ p { q . “ 2d ` d x´ x ¯ 1 In particular, takingd 1, we wee that Ω ˘ 2 . TakeX 2 andg d d for alld. For both sequences, “ 1 o 1 | | „ “ p q “ the level of distribution isx ´ p q (pretty much the best range that one can hope for), and for both sieve problems we have 2e γ V z ´ . p q„ logx However, numbersn x that have no prime factor ?x are prime or 1, thus ď ď x 2X S `, ?x 1, whileS ´, ?x . pA q “ pA q „ logx “ logx

Thus, although the sieve inputs (quantityX and bounds onA d) are identical in both problems, the truth is very different. In order to distinguish primes from integers with 2 prime factors, say, further inputs into the sieve are required. Sieve procedures which include as hypotheses bilinear sum bounds forΩ have proven to be very successful in detecting primes, e.g. Friedlander-Iwaniec [18] and Harman [21].

1.8. The small sieve. Viggo Brun’s fundamental idea was to replace the huge sum on the right side of (1.2) with a sum over a much smaller range ofd, at the expense of replacing the equality in (1.2) with an inequality. In general, a sieve (or small sieve) is a sequenceλ λ supported on squarefree integersd “p dq which replacesµ d in (1.2). We will suppose that eitherλ λ ` λ` satisﬁes p q “ “ p d q lambda+ (λ ) λ` 1, λ` 0 m 1 ` 1 “ d ě p ą q d m ÿ| or thatλ λ ´ λ´ satisﬁes “ “ p d q lambda- (λ ) λ´ 1, λ´ 0 m 1 . ´ 1 “ d ď p ą q d m ÿ| We callλ ` an upper bound sieve andλ ´ a lower bound sieve, which comes from the following easy result: SIEVE METHODS LECTURE NOTES, SPRING 2020 9

Lemma 1.1. Forz 2 letP z denote the set of squarefree positive integers, divisible only by primes ě p q p z. Assume(λ ) and(λ ). Then, for any sieve problem and andz 2, ď ` ´ ě S-lambda (1.6) λÁ S , z λÀ . d d ď pA qď d d d P z d P z Pÿp q Pÿp q Further, for any multiplicative functiong such that0 g p 1 for eachp z, we have ď p qă ď lambda-g (1.7) λ´g d 1 g p λ`g d . d p qď p ´ p qqď d p q d P z p z d P z Pÿp q źď Pÿp q Proof. For anyω Ω, let P mω p. “ p z ωźď p PA Then, by (λ`), � S , z MP m ω 1 M E mω 1 pA q“ p “ q “ ¨ “ M E λ` λÀd, ď ¨ d “ d d mω d P z ÿ| Pÿp q and similarly by (λ´),

S , z M E λ´ λ´Ad. pA qě ¨ d “ d d mω d P z ÿ| Pÿp q The claim (1.7) is a special case, where our probability space has independent events p such thatP p A A “ g p , so thatS , z M p z 1 g p andA d Mg d . � p q pA q“ ď p ´ p qq “ p q The major goal of sieve methodsś is to construct good sieve parametersλ ˘, with small support and with the sums in (1.6) mimickingS , z as closely as possible. With this notation and (r), we have pA q Alamgr (1.8) λ A X g d λ λ r . d d “ p q d ` d d d P z d P z d P z Pÿp q Pÿp q Pÿp q Asλ is a replacement forµ d , it is reasonable to suppose (if we have constructed our sieve well) that d p q g d λ g d µ d 1 g p V z . p q d « p q p q “ p ´ p qq “ p q d P z d P z p z Pÿp q Pÿp q źď 1.9. Legendre’s sieve, general version. thm:Legendre Theorem 1.2. Let be a sieve problem, assume(g),(r) and adopt notation(V). Then 2 A Legendre (1.9) S , z XV z O r . pA q“ p q` | d| ˆ d P z ˙ Pÿp q Proof. Trivially, the weightsλ ˘ µ d satisfy (λ ) and (λ ) with equality. By (1.6), d “ p q ` ´

S , z X µ d g d µ d rd XV z O rd .� pA q“ p q p q ` p q “ p q` | | d P z d P z ˆ d P z ˙ Pÿp q Pÿp q Pÿp q 2 Recall thatr d is supported on squarefree integersd. Thus, for any sum involvingr d, the sums is restricted to squarefree integers. 10 KEVIN FORD

π z Although this sieve suffers from the large number,2 p q , of remainder summands, it is useful in situations where the “densities”g p are rather small on average, and so the product on the right hand side of (1.9) p q captures the true behavior of set of interest for relatively smallz. A prominent example of the use of Legendre’s sieve was given by C. Hooley [24], who deduced Artin’s primitive root conjecture from the Generalized Riemann Hypothesis for Dedekind zeta functions of certain numberﬁelds. Details will be given later in Section 2.2.5. sec:HL 1.10. The primek-tuples conjecture. Much of sieve theory has been driven by attempts to prove special

cases of the general Primek-tuples Conjecture. The setup is aﬁnite collectionf 1, . . . , fk of nonconstant irreducible polynomials inm variables, with integer coefﬁcients. For each primep, let

ρ p # x x 1, . . . , xm modp:f 1 x f x 0 modp . p q “ t “p q p q ¨ ¨ ¨ kp q ” p qu m The collectionf 1, . . . , f are said to be admissible ifρ p p for allp. k p qă conj:ktuple Conjecture 1.3 (General Primek-tuples Conjecture). Letf 1, . . . , fk be an admissible collection of polynomials fromZ m toZ m. Then S xm prime-k-tuples-conj (1.10)# 0 x i x 1 i m :f 1 x , . . . , f x are all prime , t ď ă p ď ď q p q kp q u„ k logx k i 1 deg fi “ p q p q where deg fi is the total degree off i, and ś p q k ρ p 1 ´ S S f 1, . . . , fk 1 pmq 1 “ p q “ p ´ p ´ p ź ˆ ˙ˆ ˙ is the so-called singular series associated withf 1, . . . , fk.

One can show thatρ p p m 1 isk on average and this implies that the inﬁnite product converges (details p q{ ´ below in the case of univariate polynomials). This conjecture subsumes a large number of conjectures that have been made over time, in various degrees of generality. We mention here the conjectures of Bunyakowsky [3], Dickson [8], Hardy-Littlewood [22], Schinzel [31], and Bateman-Horn [2]. Some special cases.

Primes in an arithmetic progression.m 1,k 1,f 1 n qn a, where a, q 1. Dirichlet ‚ “ “ p q “ ` p q“ proved [9] in 1837 that there are inﬁnitely manyn withf 1 n prime; the Prime Number Theorem p q for arithmetic progressions (de la Valee´ Poissin, 1896) gives the full asymptotic (1.10). 1 Twin primes.m 1,k 2,f 1 n n,f 2 n n 2. HereS C: 2 p 2 1 p 1 2 ‚ “ “ p q “ p q “ ` “ “ ą p ´ p ´ q q “ 1.32... is the “twin prime constant”. ś “Sexy” primes.m 1,k 2,f 1 n n,f 2 n n 6. HereS 2C sinceρ 2 ρ 3 1 ‚ “ “ p q “ p q “ ` “ p q “ p q “ andρ p 2 forp 3. That is, there are twice as many “sexy primes” as twin primes. p q “ ą Sophie Germain primes.m 1,k 2,f 1 n n,f 2 n 2n 1. ‚ 2 “ “ p q “ 2 p q “ ` Primes of formn 1.m 1,k 1,f 1 n n 1. ‚ ` “ “ p q “ ` SIEVE METHODS LECTURE NOTES, SPRING 2020 11

k-term arithmetic progressions of primes.m 2, formsn 1, n1 n2, n1 2n2, . . . , n1 k 1 n2. ‚ “ ` ` `p ´ q Whenk 3, the asymptotic in Conjecture 1.3 was essentially proved by Vinogradov in 1937. Balog “ [1] extended this to include many other collections of forms withm 2. Green and Tao [11] showed ě that there are inﬁnitely manyk-term arithmetic progressions of primes for anyk 4, and this was ě extended by Green and Tao and by Green, Tao and Ziegler [12, 13, 14, 15] in 2010–12 to prove the full asymptotic in (1.10). Their theory extends to many other collections of linear forms when m 2. ě 2 2 2 2 Primes of the formx y .m 2,k 1,f 1 x, y x y . Fermat showed that every prime ‚ ` “ “ p q“ ` p 1 mod 4 is the sum of two squares. ” p q 2 4 2 4 Primes of the formx y .m 2,k 1,f 1 x, y x y . The inﬁnitude of such primes, and ‚ ` “ “ p q“ ` in fact the full asymptotic (1.10), is a celebrated theorem of Friedlander and Iwaniec [17] in 1998.

The only case of (1.10) which is known whenm 1 is the single linearf 1 case. In fact, other than the “ casek 1 andf 1 linear, it is not known a single speciﬁc example of a set of forms for which there are “ inﬁnitely manyn Z making all off i n simultaneously prime. P p q There is a relatively easy heuristic for (1.10). According to the Prime Number Theorem, a randomly 1 chosen integer nearx has a likelihood of about logx of being prime. Assuming that thef i x behave 1 1 p q randomly, the likelihood off i x being prime should be about logf x deg f logx if eachx i 1, x . p q ip q „ p iq P r s This leads to the prediction that 1 x # 0 x i x 1 i m :f 1 x , . . . , f x are all prime . t ď ă p ď ď q p q kp q u„ k logx k i 1 deg fi “ p q p q This matches (1.10) except for the singular series factor. Implicit in theś “randomness” hypothesis is the k assumption that for any primep, the likelihood that each off i x is coprime top is about 1 1 p . This p q p ´ { q is not correct, however, and ifx is chosen randomly modulop, then the likelihood that each of thef i x is p q coprime top is exactly1 ρ p p m. Thus, in our heuristic we should insert a correction factor ´ p q{ k ρ p 1 ´ 1 p q 1 . ´ pm ´ p ˆ ˙ˆ ˙ Doing this for allp produces a correction factor equal toS f 1, . . . , f , and leads to the more precise p kq prediction (1.10).

1.11. Brun’s pure sieve. Brun’s sieve is based on a simple truncated version of inclusion-exclusion, due to Brun (1915). incl-excl Lemma 1.4 (Inclusion-exclusion). Letu be a non-negative integer. Then, for anyk N, P k 8 r u r u k 1 u 1 �u 0 u 1 1 1 ` ´ , “ p q “ p´ q r “ p´ q r ` p´ q k r 0 ˆ ˙ r 0 ˆ ˙ ˆ ˙ ÿ“ ÿ“ 1 where we set ´ 0 for allj. j “ ` ˘ 12 KEVIN FORD

Proof. We may assume thatu 1, as the statement is trivial whenu 0. Theﬁrst equality is trivial from ě “ the binomial theorem. For the second, whenu 1 we have ě 8 r u 8 r u 1 u 1 k 1 u 1 1 1 ´ ´ 1 ` ´ .� p´ q r “ p´ q r 1 ` r “ p´ q k r k 1 ˆ ˙ r k 1 „ˆ ˙ ˆ ˙ ˆ ˙ “ÿ` “ÿ` ´ thm:Brunpure Theorem 1.5 (Brun’s pure sieve). Letk be a nonnegative integer and deﬁne

µ d ifω d k λd p q p qď . “ #0 otherwise;

Thenλ satisﬁes(λ `) ifk is even and(λ ´) ifk is odd. Thus, ifk e is even andk o is odd, then for any sieve problem and anyz 2 we have A ě Brun-1 (1.11) µ d A S , z µ d A . p q d ď pA qď p q d d P z d P z P p q P p q ω dÿko ω dÿke p qď p qď Proof. Letk 0. Then, by Lemma 1.4, ě k � � r ω m k 1 ω m 1 m 1 ω m 0 1 p q 1 ` W, W p q ´ 0, “ “ p q“ “ p´ q r ` p´ q “ k ě r 0 ˆ ˙ ˆ ˙ ÿ“ k r k 1 1 1 1 ` W “ p´ q ` p´ q r 0 d m “ ÿ ω ÿd| r p q“ k 1 µ d 1 ` W “ p q ` p´ q ω d k pÿqď k 1 λ 1 ` W, “ d ` p´ q d ÿ from which follows (λ`) ifk is even and (λ ´) ifk is odd. Theﬁnal claim (1.11) follows from (1.6).�

In probability theory, the inequalities in Theorem 1.5 were later rediscovered by Bonferroni, and are often referred to as the “Bonferroni inequalities”.

VW Lemma 1.6. Letz 2 and0 f p 1 for each primep z. Letk be a non-negative integer. Extendf ě ď p qď ď multiplicatively by deﬁningf d p d f p for any squarefreed withd P z . Then p q “ | p q P p q śµ d f d 1 f p 1 kW, p q p q “ p ´ p qq ` p´ q d P z p z ď ωPÿd p kq ź p qď where k 1 1 ` 0 W f d f p . ď ď p qď k 1 ! p q d P z p z p ` q ˆ ď ˙ ω dPÿpk q 1 ÿ p q“ ` SIEVE METHODS LECTURE NOTES, SPRING 2020 13

Proof. By Theorem 1.5 and (1.7),

k 1 1 f p µ d f d 1 ` W,0 W f d . p ´ p qq “ p q p q ` p´ q ď ď p q p z ω d k ω d k 1 źď pÿqď p q“ÿ ` Theﬁnal inequality forW comes from expanding the k 1 -fold sum (this is an old Erd os˝ trick).� p ` q Brun-asymp Theorem 1.7. Let be a sieve problem, assume(g) and(r). Then A 3 2 S , z XV z O XV z { O r . pA q“ p q` p p q q ` | d| ˆ d z4 log 1 V z 1 ˙ ď ÿp { p qq` whereV z is given by(V). p q 1 Proof. AssumeV z 0 otherwise there is nothing to prove. Now apply Theorem 1.5 withk 4 log V z , p qą “ p q followed by an application of Lemma 1.6. This gives Y ]

S , z µ d A O A pA q“ p q d ` d d P z ˆ d P z ˙ ωPÿd p kq ω dPÿpk q 1 p qď p q“ ` X µ d g d O X g d R “ p q p q ` p q ` d P z ˆ d P z ˙ ωPÿd p kq ω dPÿpk q 1 p qď p q“ ` k 1 1 ` XV z O X g p R, “ p q` k 1 ! p z p q ` ˆ p ` q ˆ ÿď ˙ ˙ where R r . | |ď | d| d P z ω dPÿpk q 1 p qď ` Lastly, by our deﬁnition ofk, k 1 k 1 1 ` 1 ` g p log 1 g p k 1 ! p z p q ď k 1 ! p z ´ p ´ p qq p ` q ˆ ÿď ˙ p ` q ˆ ÿď ˙ k 1 1 1 ` log UkV-Brun “ k 1 ! V z (1.12) p ` q ˆ p q ˙ k 1 1 ` e log V z p q ď k 1 ˜ ` ¸ k 1 4 log 4 4 3 2 e 4 ` V z ´ V z { . ďp { q ! p q ď p q This completes the proof. �

Remarks. We have imposed virtually no hypotheses on the sieve problem in this theorem. In ap- A plications, one has typicallyV z logX κ for someﬁxedκ, and thus we have applied (1.11) with p q—p q ´ k 4κ log logX. Typically in the sum in identity (1.2) and the sums in Brun’s inequalities (1.11), the « summands corresponding tod X are negligible (or identically zero). Hardy and Ramanujan [23] in 1917 ą 14 KEVIN FORD showed that most integersd X have about log logX prime factors, and thus it is natural to choosek ď somewhat larger than log logX in order to capture the bulk of the sum.

1.11.1. Example: twin primes. As before, letΩ 1, x Z, p k Ω:p k k 2 . Take “r s X A “ t P | p ` qu ρ p g p p q,ρ 2 1,ρ p 2 p 2 p q “ p p q “ p q “ p ą q and then, for squarefreed we have r ρ d τ d . | d|ď p qď p q By Mertens’ estimate (cf., (1.5)),V z c logz 2 for some constantc. Take p q„ p q ´ 1 z x 16 log logx , “ so that in the summation of the error termsr d we have

4 log 1 V z 1 8 log logz O 1 1 2 o 1 z p { p qq` z ` p q x { ` p q. “ ! Thus, 1 2 o 1 rd τ d x { ` p q. | |ď 4 log 1 V z 1 p q ! d P z d z p { p qq` ω d 4 logPÿ1p Vq z 1 ď ÿ p qď p { p qq` By Theorem 1.5, cx log logx 2 S , z XV z 256cx . pA q„ p q„ log2 z „ logx ˆ ˙ AsS , z # z k x:k,k 2 both prime , we see that pA qě t ă ď ` u log logx 2 # k x:k,k 2 both prime x , t ď ` u ! logx ˆ ˙ which misses the conjectured order by a factor log logx 2. Applying partial summation gives an immediate p q corollary.

Corollary 1.8 (Brun [6], 1919). We have 1 . p ă8 p:p,p 2 prime `ÿ Remarks. Applying Theorem 1.5 to the sieve of Eratosthenes yieldsπ x x log logx , missing the true p q ! logx order ofπ x by a factor log logx. p q Brun later [7] gave a much more complicated version of his sieve, where the simple truncation (1.11) is replaced by sieves where one considers the summation over integersd with restricted prime factors of various sizes. A greatly simpliﬁed version of this idea was found by Hooley, and which will be the subject of the following section.