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 for k properties this produces 2k summands which is usually too much to effectively deal with. A sieve is a procedure to estimate the number of “desirable” elements of the set using kOp1q 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. τpnq is the number of positive divisors of n !pnq is the number of distinct prime factors of n Ωpnq is the number of prime factors of n counted with multiplicity µpnq is the Mobius’s¨ function; µpnq “ p´1q!pnq if n is squarefree and µpnq “ 0 otherwise. P `pnq is the largest prime factor of n; P `p1q “ 0 by convention P ´pnq is the smallest prime factor of n; P ´p1q “ 8 by convention Ppzq is the set of positive squarefree integers composed only of primes ¤ z Λpnq denotes the von Mangoldt function 1X is the indicator function of the statement X or of the set X the symbol p, with or without subscripts, always denotes a prime πpx; q; aq is the number of primes p ¤ x in the progression a mod q. P denotes probability and E expectation logk x is the k-th iterate of the logarithm of x 1.2. General sieve setup. A sieve problem A is a probability space pΩ; F; Pq, together with a “total mass” quantity M and events Ap, one for each prime p. The “sifting function” is SpA; zq “ M ¨ Ppnot Ap; @p ¤ zq: Oftentimes Ω will be a finite set of integers, with Ppn P Ωq “ 1{|Ω| for each n P Ω (the uniform probability measure), M “ |Ω| and Ap is the event that p|n, that is, Ap “ tn P Ω: p|nu. This is the standard “small 1 2 KEVIN FORD sieve” problem, where SpA; zq is the number of n P Ω with p - n for all p ¤ z; these are called the “unsifted 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 “ r1; xs X , uniform probabilities on Ω, M “ |Ω| “ txu, and ? N ? A “ tn P Ω: p|nu. Then SpA; xq “ πpxq ´ πp xq ` 1, as the unsifted elements are the primes p ? in p x; xs together with the number 1. Twin primes. Ω 1; x , A n Ω: p n n 2 . The unsifted numbers are numbers n ‚ “ r s X N p “ t P | p ` qu ? ? such that npn ` 2q has no prime factor ¤ z. In particular, SpA; x ` 2q counts k P p x ` 2; xs for which both k and k ` 2 are prime. Equivalently, Ap is the set of n that avoid the residue classes 0 mod p and ´2 mod p. ‚ Twin primes, weighted version. Ω “ r2; xs X N, and Λpn ` 2q pn P Ωq “ ;M “ Λpn ` 2q; P M 2¤n¤x ¸ Ap “ tn P Ω: p|nu. Here M “ 2¤n¤x Λpn ` 2q „ x by the Prime Number Theorem, and ? SpA; xq “ ř Λpn ` 2q “ Λpp ` 2q ? 2¤n¤x? xăp¤x P ´p¸nq¡ x ¸ “ logpp ` 2q ` Opx1{2q; ? xăp¤x p`2¸prime the error term coming from terms p ` 2 “ qb where q is prime and b ¥ 2. 1 ‚ Prime tuples. Let a1; ¨ ¨ ¨ ; ak P N and b1; : : : ; bk P Z. Put Ω “ r1; xs X N, and Ap “ tn P Ω: p|pa1n ` b1q ¨ ¨ ¨ pakn ` bkqu: ? For an appropriate c ¡ 0, which depends on a1; a2; : : : ; ak; bk, SpA; c xq counts those n ¤ x for which a1n ` b1; : : : ; akn ` bk are simultaneously prime. ‚ Prime values of a polynomial. Let f : Z Ñ Z be an irreducible polynomial of degree h ¥ 1, put h Ω “ r1; xs X Z, Ap “? tn P Ω: p|fpnqu. Let C be large enough so that |hpnq| ¤ Cn for all n ¥ 1. Then, as before, SpA; Cxhq captures values of n for which fpnq is prime. ‚ Goldbach’s problem. Let N be an even, positive integer,? put Ω “ t1; 2;:::;N?´ 1u, M “ |Ω| “ N ´ 1, and Ap “ tk P Ω: p|kpN ´ kqu. Then S?pA; Nq counts numbers k P p N;Ns for which both k and N ´ k are prime. In particular, SpA; Nq ¡ 0 implies that N is the sum of two primes. If one shows this for all N ¥ 4, one deduces Goldbach’s Conjecture. ‚ Primes in an arithmetic progression. Fix coprime positive integers a and q, let Ω “ t1 ¤ n ¤ x : n ” a pmod qqu; ? ? Ap “ tn P Ω: p|nu. Then SpA; xq captures primes in p x; xs that are in the arithmetic progres- sion a mod q. 1Unless otherwise specified, from now on whenever Ω is a finite set, the probability measure on Ω will be the uniform measure, and M will be the number of elements of Ω. SIEVE METHODS LECTURE NOTES, SPRING 2020 3 ‚ Sums of two squares. Ω “ r1; xs X Z, tn P Ω: pe}n for some odd eu p ” 3 pmod 4q Ap “ #H otherwise. Then SpA; xq counts integers n ¤ x, for which we don’t have pe}n for any prime p ” 3 pmod 4q and odd exponent e. That is, SpA; xq is the number of integers n ¤ x which are the sum of two squares. ‚ Sieve by multiple residue classes. Let M; N be two integers, Ω “ tN ` 1;M ` 2;:::;N ` Mu and for each prime p let Ip be some subset (possibly empty) of the residue classes modulo p. Put Ap “ tn P Ω: n mod p R Ipu: Here SpA; zq counts the integers in pN; M ` Ns avoiding all the residue classes Ip for primes p ¤ z. If |Ip| is bounded or bounded on average, then this is a very general sieving problem of “small sieve” type, whereas if |Ip| is unbounded on average, the problem falls under the umbrella of the “large sieve”. The case of prime values of a polynomial, see above, is a special case with Ip “ tn P Z{pZ : fpnq ” 0 pmod pqu: k k ‚ Multivariate polynomial sieve. Let F pxq : Z Ñ Z be a multivariate polynomial of x “ k px1; : : : ; xkq, take any finite Ω P Z , and Ap “ tx P Ω: p|F pxqu. Then SpA; zq counts x P Ω for which F pxq has no prime factor p ¤ z. ‚ The square-free sieve. Let Ω be a finite set of integers and for each prime p let Ap “ tn P Ω: p2|nu. Then SpA; zq, with an appropriately large z, will count the elements of Ω which are squarefree. A famous application is for squarefree values of a polynomial, e.g. Ω “ tfpnq : 1 ¤ n ¤ xu, where f is an irreducible polynomial. One can similarly set up a k-free sieve problem. ‚ Elliptic curve sieve. Fix and elliptic curve E over Q Let Ω be the set of primes q ¤ x. Let A “ tq P Ω: p|#E{ u, where E{ is the reduction of E modulo q. It is known that #E{ ¤ p ? Fq ? Fq Fq q ` 1 ` 2 q, and thus SpA; 2 xq counts those q for which #E{Fq is prime. If d is a square-free integer composed only of primes in P, define (1.1) Ad “ Ap;Ad “ M ¨ PpAdq: p d £| In particular, A1 “ M. In the case where Ω is a finite set of integers with uniform measure, M “ |Ω|, and Ap “ tn P Ω: p|nu, Ad counts the number of n P Ω divisible by d. In this notation, inclusion-exclusion gives (1.2) SpA; zq “ µpdqAd: P ` d z ¸p q¤ 1.3. Legendre’s sieve for primes (Legendre, 1808). Let Ω “ r1; xs X Z, Ap “ tn P Ω: p|nu. Then SpA; zq counts the positive integers n ¤ x with no prime factor ¤ z; in particular, this includes all of the primes between z and x. Also, for each aquarefree d, Ad “ tn P Ω: d|nu and thus Ad “ |Ad| “ tx{du “ x{d ` Op1q: 4 KEVIN FORD By (1.2), x πpxq ´ πpzq ¤ SpA; zq “ µpdq ` Op1q d P ` d z ¸p q¤ ´ ¯ µpdq “ x ` OpτpP qq d P ` d z ¸p q¤ 1 “ x 1 ´ ` O 2πpzq : p p¤z ¹ ˆ ˙ ´ ¯ Taking z “ log x, and using the crude bound πpzq ¤ z together with Mertens’ bound, we conclude that x x πpxq ¤ z ` e´γ ` op1q ` Op2log xq “ O : log z log x ˆ 2 ˙ ` ˘ Remark: We can do better by observing that in fact Ad “ 0 for d ¡ x (cf.