SIEVE METHODS LECTURE NOTES, SPRING 2020 KEVIN FORD 1. BASIC SIEVE METHODS AND APPLICATIONS 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 .