This page intentionally left blank CAMBRIDGE TRACTS IN MATHEMATICS General Editors B BOLLOBAS, W FULTON, A KATOK, F KIRWAN, P SARNAK, B SIMON, B TOTARO 177 A Higher-Dimensional Sieve Method A Higher-Dimensional Sieve Method HAROLD G DIAMOND H HALBERSTAM With Procedures for Computing Sieve Functions WILLIAM F GALWAY CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521894876 © H Diamond, H Halberstam andW Galway 2008 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2008 ISBN-13 978-0-511-43731-1 eBook (EBL) ISBN-13 978-0-521-89487-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents List of Illustrations List of Tables Preface Notation Part I Sieves page xi xiii xv xvii 1 Introduction 1.1 The sieve problem 1.2 Some basic hypotheses 1.3 Prime g-tuples 1.4 The n(n) condition 1.5 Notes on Chapter 3 11 Selberg's sieve method 2.1 Improving the Eratosthenes-Legendre sieve 2.2 A new parameter 2.3 Notes on Chapter 13 13 14 17 Combinatorial foundations 3.1 The fundamental sieve identity 3.2 Efficacy of the Selberg sieve 3.3 Multiplicative structure of modifying functions 3.4 Notation: V, S(A, V, z), and V 3.5 Notes on Chapter 19 19 22 25 26 27 The 4.1 4.2 4.3 29 29 33 42 Fundamental Lemma A start: an asymptotic formula for S(A.q, "P, z) A lower bound for S(Aq, V, z) Notes on Chapter viii Contents Selberg's sieve method (continued) 5.1 A lower bound for G?(£, z) 5.2 Asymptotics for G*(£,z) 5.3 The j and a functions 5.4 Prime values of polynomials 5.5 Notes on Chapter 43 43 52 56 64 66 Combinatorial foundations (continued) 6.1 Statement of the main analytic theorem 6.2 The S(x) functions 6.3 The "linear" case K — 6.4 The cases K > 6.5 Notes on Chapter 67 67 70 71 73 79 The 7.1 7.2 7.3 7.4 7.5 81 81 85 88 92 95 An application of the linear sieve 8.1 Toward the twin prime conjecture 8.2 Notes on Chapter 97 97 102 A sieve method for K > 9.1 The main theorem and start of the proof 9.2 The S2i and S22 sums 9.3 Bounds on S ± 9.4 Completion of the proof of Theorem 9.1 9.5 Notes on Chapter 103 103 107 110 120 122 10 Some applications of Theorem 9.1 10.1 A Mertens-type approximation 10.2 The sieve setup and examples 125 125 129 11 A weighted sieve method 11.1 Introduction and additional conditions 11.2 A set of weights 11.3 Arithmetic interpretation 11.4 A simple estimate 11.5 Products of irreducible polynomials 11.6 Polynomials at prime arguments 11.7 Other weights 11.8 Notes on Chapter 11 135 135 137 140 144 147 149 150 151 case K = 1: the linear sieve The theorem and first steps Bounds for V S ± : the set-up Bounds for V £ * : conclusion Completion of the proof of Theorem 7.1 Notes on Chapter 252 Procedures for computing sieve functions Equation (A1.44) has a family of solutions zm, m £ Z , satisfying (A1.45) zm = 2nim + log(«) - log(l + K - uzm), 5Rzm < Note that Szo = We can show that Sz m = 2irm + O(l), with an absolute, and small, O-constant, and that 5Rzm = — log(|m|) + 0(1) for m ^> K/U For each m, zm is readily approximated by starting with the initial approximation z = — log(l + 1/K), and then iterating the map z H> 2irim + log(«) — log(l + K — uz) until z fails to change by more than our desired error tolerance We evaluate the integral of (A1.42) along the path {zo, • • •, zM}, where the terminal saddle point, zM, is chosen to bound the truncation error arising from using a finite path We not ensure that the path passes through these saddle points in the "optimal" direction as given by equation (A1.28) This casual approach is justified by the remarks following equation (A1.45), which suggest that the saddle points lie along a nearly vertical path; so in going straight from one saddle point to the next the direction followed is not too far off the mark Before discussing the choice of zM, we turn to evaluation of the integral in equation (A1.41), where the path lies to the right of the singularity at the origin (see Figure Al.ll) This proceeds almost precisely as when evaluating (A1.42), but with a different choice for the first few points defining the path In addition to the saddle points zm, G(z) also has a saddle point at z = z+, where z+ denotes the positive real solution of equation (A1.44) We can easily show that 1/u < z+ < (K + l)/u, and we compute z+ by applying Mathematical FindRoot operator to equation (A1.44) with the initial approximation of z = (K + l)/u Letting z0+ := max(10, z+), we use the three points Zf;+ e z0+, u ^ ô ^ /, as the first points along the path used when evaluating (A1.41) (We arbitrarily bound z0+ by 10 to prevent the path from lying too far to the right when u is very near 0.) Letting MQ := [zo+l /(2TT), we use {zMo, , z M } as the remaining points along our path To summarize, we use the path {z0+, z1+, z2+, ZMQ, , zM} when evaluating (A1.41) To find our terminal point zM, without any rigorous error analysis we A 1.8 Computing jK(u) 253 10 -2 Fig A l l l log 10 \G(x + iy)\, where G(z) is the integrand appearing in equation (A1.41), contour interval is 0.5, n = 9/2, u = observe that \G(zm)\ decreases nicely as m increases, and that f>zm+ioo G(z) di \G(zm)\\zm On this basis, for a given truncation-error tolerance £ we simply choose M to satisfy \G(zM)\ < e (Note that M = in Figure A1.10, while M = in Figure Al.ll.) To choose between equations (A1.41) and (A1.42) as approximations for jK(u), we recall that (A1.46) (A1.47) jK(u) = —jK(u) = du +1), < u < 1, njK(u)-KJK(u-l) From equation (A1.46) we see that jK(u) is very small when u < and K is large In Chapter 14 we showed that jK(u) increases to as u —> oo, and equation (14.41) implies that jK(n) TH 1/2 254 Procedures for computing sieve functions It follows that when u is much smaller than K then jK (u) is small—very small when K is large In this case the two terms in equation (A1.42) must nearly cancel, and we would need to evaluate the integral in (A1.42) to very high accuracy to get an acceptable relative-error bound in our approximation of jK (u) On the other hand, when u is much larger than n then the integral term in (A1.42) is near 0, and need only be found with modest accuracy to estimate jK(u) accurately For these reasons, we use equation (A1.41) when K < u and equation (A1.42) when n > u Since it is readily computed, we always use equation (A1.46) to find jK(u) when u < 1, except when, for testing purposes, we wish to compare the results from our contour integral representations against (A1.46) Instead of our contour integral formulas, when u > we may use NDSolve to solve equation (A1.47): the defining delay differential equation for jK(u) Since we save the InterpolatingFunction returned by NDSolve, this approach has the advantage that, once jK{u\) has been computed, given < u < u\ we can find jK(u) more rapidly than by evaluating a contour integral However, we find that the solution returned by NDSolve tends to drift away from the correct value when u > K, particularly when K is large For this reason, to balance the goals of speed and accuracy, only when u £ Z we use a contour integral formula to find jK(u) Otherwise, we apply NDSolve to compute jK(u) over the interval [u] < u < [u] + Note that we use the contour integral value for jK([u]), and this initial condition for NDSolve reduces the potential for drift in our solution A1.9 Computing aK and j3K To compute aK and (3K we use Mathematical FindRoot operator to solve the system of equations (A1.48) (P,p)_ ( e (a)-2 = 0, al-2K(Q,q)K(a) = 0, where (P,p)_K(u) and (Q,q)R(u) are approximated by applying Mathematical NIntegrate operator to their defining expressions as implied by equation (A1.3) FindRoot requires initial approximations to aK and /3K, which we interpolate (or extrapolate) from an InterpolatingFunction 255 Al.10 Weighted-sieve computations which was "bootstrapped" from a few values calculated using the software developed by Ferrell Wheeler [Whe88] Equation (A1.48) includes the scaling factor a1~2K since FindRoot is not well suited to solving two equations in two unknowns when the magnitude of acceptable error differs greatly between the two equations For example, with K = 10, we calculate that (P,p)_ B K) - ô 5.32463 1(T13, a i - K ( Q , < K ) « 2.99870 -lO" 15 , while the latter inner product when unsealed is computed as 8.97906 • 1013 (Q,q)K(aK)^ Al.10 Weighted-sieve computations In Chapter 11 we used a weighted sieve to put lower bounds on r for which almost-prime numbers, with at most r prime factors, occur frequently in a sequence A In Theorem 11.1 we introduced the function N{u, v; K, /u.o, T) which serves as such a lower bound To implement this function, we change variables in the integral of equation (11.16), which defines N(u,v; K,/IO,T) Letting s — i > 1/t gives N(U,V;K,HO,T) ( A1 - 9) 11 + + K // r'u , / V u \ dt v ) t F KF{ KT{Tv-t)(l t)— v t ) ( l t ) We use NIntegrate to approximate the right side of (A1.49), taking care not to integrate across the points t where TV - t e Z U (aK + Z) U {/3K + Z), at which the integrand has bad differentiability properties (See the discussion at the end of Section A 1.4.) Of course, to get the best possible lower bound in Theorem 11.1, we want to approximate the minimal value achieved by N{u, v; K, fio, r) when u and v are free to vary (subject to the conditions imposed by that theorem) That is, we want to approximate (A1.50) Nmin{K,fi0,T) := N{u,v; K, /XO,T) V>L/T 1/T• oo (See Figure A1.12.) Since Mathematica is not well-suited to search nonrectangular regions for a root or minimum, we reparametrize u in terms of a variable v, setting (A1.51) u = u{v) := — T so that v — log(« — 1/T) — log(« — 1/r) Figure A1.13 shows a contour plot oiN{u{y),v\K,\iQ,T) With this change of variable, we can restate equation (A1.50) as (A1.52) Nmin(K,fi0,T) = V>(3K/T 0