THE ANALYTIC LARGE SIEVE

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1 THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig the least quadatic o-esidue. I this lectue we will deive the aalytic lage sieve. Fo moe discussio of the lage sieve I ecommed the excellet suvey aticle witte by Motgomey []. Befoe statig the mai esult let us itoduce the followig otatio. Give a eal umbe α defie the eaest distace fom α to a itege as α := mi α. Z Defiitio. A subset of eal umbes {α } is called δ-spaced if fo s α α s δ. Also, fo complex umbes a wite L(α) = a e(α). Theoem 1.1 (The aalytic lage sieve 1). Let {α } be δ-spaced the L(α ) (X + 1/δ 1) a. Specializig the poits α gives the followig Coollay 1. (The aalytic lage sieve ). We have ( ) a L ( X + Q 1 ) q q Q a (mod q) Poof. Coside the set of poits 1 q Q 1 a<q gcd(a,q)=1 {a/q}. a. It suffices to show this set of poits is 1/Q -space. To see this obseve that a 1 a q 1 q = a 1 q a q 1 q 1 q 1 1 q 1 q Q. Date: Jue 11,

2 THE ANALYTIC LAGE SIEVE. Duality As a fist attempt, let us ty to diectly estimate the secod momet of L(α ). We get that L(α ) = a m a e(α (m )) M<m, M+X Howeve, at this stage we do ot have eough ifomatio about the poits {α } to poceed futhe. We ow poceed i a diffeet diectio. Coside a liea map Θ : C X C, which ca be expessed as Θ = {θ(, )} M+X, θ(, ) = e(α ) ad α = (a M+1,..., a M+X ) Θα = a e(α ). Ou goal is to show that =M+1,=1. Θα (X + 1/δ 1) α. Let us ow ecall that the om of Θ is give by Θ = if{c : Θα c α fo all α}. Obseve that fo The adjoit of Θ which is deoted by Θ is defied via the fomula Θα, β = α, Θ β ad such a opeato exists by the iesz epesetatio theoem. I ou settig Θ is just the cojugate taspose, which we will deote by Θ. Obseve that fo β = (b 1,..., b ) Θ β = b e( α ). ecall the followig esult which we state as Θ = Θ, Popositio.1 (Duality). Suppose that thee exists such that fo b C (1) b e(α ) b. The fo a C () The covese holds as well. a e(α ) a.

3 THE ANALYTIC LAGE SIEVE 3 emak. Duality is a key compoet of the poof that we will give. It is atual to wat to give a asymptotic fo L(α ) ad oe might hope that if a has exta stuctue say fo example a = τ() this might be possible. Howeve, this caot be doe if we use duality sice i ode to establish (1) we eed to kow that () holds fo moe geeal b ad we lose the exta stuctue of the a. By othe methods, ecet wok of Coey, Iwaiec, ad Soudaaaja [1] gives a asymptotic lage sieve i a special, ad impotat, settig. Poof. This is a special istace of the fact that the om of a liea opeato is equal to the om of its adjoit. Fo claity we will give a diect poof. We will show (1) implies () the covese follows fom a simila agumet, which we will omit. Defie Ψ = a b e(α ) ad take b = a m e(mα ). m So that Ψ = a a m e(( m)α ) =,m Applyig Cauchy-Schwaz we also have Ψ = a b e(α ) a b e(α ) a b. a e(α ). Now obseve that b = a e(mα ) = Ψ. Combiig estimates gives a e(mα ) a as claimed.

4 4 THE ANALYTIC LAGE SIEVE 3. A poblem i Fouie aalysis 3.1. Backgoud o Fouie aalysis. Fo f L 1 () defie the Fouie tasfom of f by f(ξ) = f(x)e( xξ) dx. ecall the space of Schwaz fuctios, S() is the set of smooth fuctios such that fo ay i, j 0 x i f (j) (x) i,j 1. We ow ecall some basic facts Fo f S() the mappig f f is a cotiuous ijective map of S oto itself. The Fouie tasfom is a isomety o S(), i.e. f = f. ad Placheel s theoem exteds this isomety to f L 1 L. A special case of the ucetaity piciple is that both f, f caot both have extemely apid decay. I paticula, both caot be compactly suppoted. Fo f S() the Poisso summatio fomula states f(m). Z f() = m Z I fact the above fomula is valid fo f such that f(x), f(x) (1 + x ) 1 ε. 3.. A extemal poblem aisig fom the lage sieve. Fom the pevious sectio we kow that by duality it suffices to pove that fo complex umbes b b e(α ) (X + 1/δ 1) b. We ow poceed diectly to see that b e(α ) = b b s 1 [1,X] ( + M)e((α α s )).,s The sum o the ight-had side has bee witte i a suggestive way. We would like to apply the Poisso summatio fomula f() = f(m). Z m Z Z to the fuctio f(x) = 1 [1,X] ( + M)e((α α s )). Ufotuately fo us 1 [ 1,1] (ξ) = si(πξ), πξ

5 THE ANALYTIC LAGE SIEVE 5 is ot i L 1 () so applyig Poisso summatio is ot valid. Sice we ae oly coceed with uppe bouds we will ow modify the poblem by takig a smooth majoat of 1, which has a Fouie tasfom that is compactly suppoted (hece the majoat itself will ot be compactly suppoted). Notice that if F (x) 1 [1,X] (x) is a Schwaz fuctio (o is such that both F (x), F (x) (1 + x ) 1 ε ) the we get by applyig Poisso summatio b e(α ) F ( + M) b e(α ) = b b s e( mm) F (m (α α s )).,s If i additio we have that F (ξ) = 0 fo ξ δ the sice {α } is δ-spaced b e(α ) b b s e( mm) F (m (α α s )) (3),s m = F (0) m b The extemal poblem. Note that F (0) = F (x) dx. ad F (x) 1 [1,X] (x). So we have the followig extemal poblem i Fouie aalysis. Give a fuctio whose Fouie tasfom that is compactly suppot o ξ δ how small ca ( F (x) 1[1,X] (x) ) dx be? Idepedetly, Beulig ad Selbeg solved this extemal poblem. Moeove, they gave explicit costuctios of the miimizig fuctio F (x) which solves this poblem. Theoem 3.1. Let δ > 0. Thee exists a fuctio F δ (x) such that i) F δ (x) 1 [1,X] (x), ii) F δ (ξ) = 0 fo ξ δ, ( iii) Fδ (x) 1 [1,X] (x) ) dx = 1/δ. iv) F δ (x) δ,x (1 + x ). emak. Fo futhe discussio of these fuctios see sectio 0 of Selbeg s lectues [3] o sieves. Additioally, Motgomey s suvey aticle [] o the lage sieve gives a quick oveview of these fuctios ad Vaale [4] gives a moe extesive study.

6 6 THE ANALYTIC LAGE SIEVE Usig the Beulig-Selbeg fuctio i (3) gives b e(α ) F δ (0) b = (X 1 + 1/δ) Theefoe, applyig Popositio.1 we have poved a e(α ) (X + 1/δ 1) efeeces a. b. [1] Bia Coey, Heyk Iwaiec, ad Soudaaaja. Asymptotic lage sieve. Available o the axiv at axiv: [] Hugh Motgomey. The aalytic piciple of the lage sieve. Bull. Ame. Math. Soc. 84 (1978), o. 4, [3] Atle Selbeg. Collected papes. II. epit of the 1991 editio [M195844]. With a foewod by K. Chadasekhaa. Spige Collected Woks i Mathematics. Spige, Heidelbeg, 014. viii+53 pp. [4] Jeffey Vaale. Some extemal fuctios i Fouie aalysis. Bull. Ame. Math. Soc. 1 (1985), o.,