Finding Primes by Sieve Methods

Transcription

1 Proceeings of the International Congress of Mathematicians August 16-24, 1983, Warszawa D. R. HEATH-BROWN Fining Primes by Sieve Methos At the 1978 International Congress Professor Iwaniec mentione, as an application of the linear sieve, the fact that the ifference between consecutive primes p n satisfies JPii+l-fti <Pn (1) for any #> J. This result is ue to Heath-Brown an Iwaniec [3] an improves an earlier theorem of Iwaniec an Jutila [6] which covere the range # > only. These results were obtaine by combining sieve methos with the existing machinery of the Riemann Zeta-function an zero ensity estimates. The earlier methos culminate in Huxley's proof [4] that x 7 (so that (1) hols for ê > ~, a fortiori). The constant ^ is still the limit for such an asymptotic formula ; the ranges ê > ^, # > arise from lower bouns of ut(œ + œ )-7t(oo)^G(ê)- (2) Ioga? with 0 < 0(0) < 1. The purpose of this lecture is to examine recent evelopments in this circle of ieas. Iwaniec [5] an Pintz [7] inepenently have extene the range of (1) to #> [ = , but I shall not go into this; it was alreay known in 1978 that some improvement on J = 0.55 must be possible. I shall be concerne with an analysis of the sieve principles involve in such methos an with their applications ot other problems. Let us first consier the metho of Heath-Brown an Iwaniec. One starts with the sets sé = {n; œ < n < œ + œ% sé k = {n e sé-, 7c w}, an [487]

2 488 Section 3: D. K. Heath-Brown consiers the sieve function S(sé h,z) = #{ne sé k ',p\n=>p^4. Then is(sé,x k ) is essentially the number of primes in sé. The Buchstab ientity yiels S(sé, é) = #sé- %8(sé a, q), (3) where q runs over primes. Gareful use of the linear sieve, employing Iwaniee's bilinear form for the remainer sum, allows one to give upper bouns for S(sé a, q). This prouces an estimate of the form (2), vali for < # < 1. The constant <7(#) is a continuous function of #, but unfortunately satisfies G(ê) -o (#>JL), cf W <o(#<a). To get a positive constant G(ê) one takes a range Q ± < q < Q 2 in (3) an replaces the sieve upper boun for S(sé a, q) by an asymptotic formula. This improves the overall result by G f (&)œ & (logoo)" 1, where G'(&) is continuous an positive. It follows that G(ê)+G'(ê) >0 on some interval ^ <5<#<1. When & < ~ there are only certain special ranges Qi < 2 < Qz in which asymptotic formulae are at present available. It is natural to try to evelope (3) by repeating the Buchstab ientity on those terms S(sé q, q) which have not been estimate asymptotically. This woul prouce S(sé q,q) =#sé a - S(sé Qr7 qr). r<a Again one coul eal satisfactorily with certain particular ranges of q an r. The remaining terms, which now have positive sign, may either be boune below by zero or iterate further. The process is precisely the same as that use to prouce the Rosser sieve. Olearly it is important to ientify those sums ^S(sé, z) for which an asymptotic formula can be given. (Here Jj' means that runs over those integers whose prime factors p i lie in certain specifie intervals Pi < Pi ^ 2Pj. ) The metho for hanling these sums is such that if J ' 8( sé, z) can be ealt with then so can J '(# <*-2>(<a,ï)). q<z

3 Fining Primes hy Sioye Methos 489 By repeate iterations of Buchstab's formula it therefore suffices to consier J?' # ^- Vaughan's ientity may be use for the sum over p i, P i <p i^:2p i, but cannot help when P^- < OJ (1 ~ ö)/2. One shoul therefore start from the ientity OO j n>z k=l n(s) = [J(i~p~ s ), p<z with z s= a?* 1 """^2, an pick out terms with oo < n < oo + œp. The right han sie prouces ] Jc~ 1 ( l) h ~ l 8 k (sé,z), say, while the left han sie is /c essentially 7t(œ + x a ) 7c(a)). If one applies the Buchstab iteration to 8 k (sé, z) rather than S(sé,z) one always has P^ < aß-w*, so that Vaughan's ientity is unnecessary. If 3 is the set of for which ' # sf ä can be estimate accurately, the question of bouning 8(sé,z) from below may be formulate as a general sieve problem: maximize subject to e \n,üe9 [^9 n>l. Since ^ is ifficult to ientify, no iniviual case of this sieve problem has been solve. None the less, for <# <^, some fairly sharp bouns for 7t(x-\-œ & ) n(x) are available. Specifically, when \ < # < ^, one ean give asymptotic formulae for an S(*,tf) + ^#{(p li...,p 6 ); i8f(j/,«*)-ì #{(i» lf...,ft); ft^û^-w, ft.-.fterf} (5) 2>i,., P*<» 2(1 -* )/5, JPs <» 4(1 - fl)/5, A...A6 J*}. (6)

4 490 Section 3: D. K. Heath-Brown Thejse lea to with 1X ' w(fl7 + 0*)-w(0) ' logo? Ioga? 7 s / 7 \ / 7 \ 5 0,(0) lv ; =1 0j, GL(tf) 2K J = \12 /' 1440 \12 / In particular 0$) == , 0 a (f) 4= The expression (5) arises from (4), taking z = œ 1/7. For jfc > 7 we have 8 k (sé,z) =0, an for &<5 we can give asymptotic formulae for S k (sé,z). There is also part of 8 6 (sé,z) that can be ealt with, an the expression (5) reflects what is then left. For (6) the argument also.starts with (4) but is more complicate. These ieas also yiel new information.at ê = ~. We have A secon application of this circle of ieas concerns intervals (x, x + x 9 ] which "almost always" contain a prime. By this we mean that the set of œ < X for which the interval oes not contain a prime has measure o(x) as X->oo. It follows from Huxley's work [4] that if #> then there will almost always be asymptotically ^(loga?)"*" 1 primes in the interval. Using sieve methos combine with ieas from the theory of zero ensity estimates it is now known that for & >^ the interval (x, x + x ê ] almost always contains > (0-15)0* (logo?)"" 1 primes. Let us see how this comes about. Let sé k be as before. From the Buchstab formula we have 8(sé,x*)=S(sé,x*)- 2 S (*P>P) x v <p<x^ x v <a<p<x^ If ê > ^ an <p = ~ then one can give asymptotic formulae for the first two terms of (7) for almost all x, an the thir term may be boune below by zero. One can then conclue that n(x + x»)-7t(x) > iogx

5 Fining Primes by Sieve Methos 491 for almost all x. Unfortunately O(^) == It is therefore necessary to examine the iscare terms 8(sé m, q) to see what one can salvage. In fact there are substantial ranges of p, q that can be ealt with satisfactorily, to prouce a saving of at least 0.16 in O(^). Thus far I have not mentione how the asymptotic formulae referre to may be obtaine. I shall illustrate the process by consiering say. We write #{(yim-»p*)i-p<<ft<2p i,ii 1.. t p Ä 6 sé} = N, fn(*) = E P~ U > P n <p<2p n an use the fact that for y > 0, y # 1 "we have 1 T 2izi I^-»T~ B^+0 {T^ï)' < 8 > where E(y) = J (y >1) or = \(y < 1). Then M --L f «.+^-0/,M.../,(i)i +o(«). Here we choose 2* = ^""^(loga?) 2 *, so that the error term becomes negligible. In the range \t\ < (logo?) 2 * we can replace (x-\-x û ) u x u by x» a?(loga?) ="{( +össf)"- - "} with negligible error. Then, using (8) in the reverse irection, we see that the range \t\ < (log#) 27c relates N to the number of solutions of p x.-.p k e (x, x + xqogxy* 1 *]. The latter may be calculate via the prime number theorem, an prouces the main term of the asymptotic formula for N. There remains the interval (loga?) 2& < f < T, an it is here that the size an number of the P { is important. Since the relèvent contribution is (x + x^f-x** < \t\a>*~ l 9 T < *~ 1 f l/i(*). /*(*)!*> (9) (logœ) 2A!

6 492 Section 3: D. E. Heath-Brown which we wish to be 0(x(logx)~ 2k ), say. We can now use two tools from the theory of zero ensity estimates, namely the mean value theorem for Dirichlet polynomials an the Halâsz lemma. These may be applie to a number of ifferent proucts of the/^-, proucing various sets of conitions on the P i uner which (9) is sufficiently small. A further iscussion of this may be foun in Heath-Brown [2]. In conclusion I woul like to mention one other problem where sieve methos have been successfully applie, namely Diophantine approximation with primes. Let a be irrational an let fi be any real number. Vinograov showe that if e > 0 then \\ap+ß\\<p 8-1,S (10) for infinitely many primes. (Here a? is the istance from x to the nearest integer.) This was improve by Yaughan who showe that one can take jp -i (logjp) 8 on the right. In both cases the number of relèvent primes in the appropriate interval was estimate asymptotically. By applying a sieve metho in conjunction with Vaughan's estimates, Harman [1] has recently shown that one can take p~ 3110 on the right of (10). Here one uses bouns for exponential sums as oppose to the Dirichlet polynomial techniques of the previous problems. The boun p~* is beyon the scope of these methos. Eviently there are many more potential applications of sieves in locating primes, an much work remains to be one. For example, it is not yet clear that the boun p n+ i p n <P* +S is out of reach. I feel that the significance of these ieas is not yet fully appreciate, nor is their power fully exploite. References [1] Harman G-., On the Distribution of ap Moulo 1, J. Lonon Math. Soc. (2) 27 (1983), pp [2] Heath-Brown D. E., Prime Numbers in Short Intervals an a Generalize Vaughan Ientity, öan. J. Math. 34 (1982), pp [3] Heath-Brown D. K. an Iwaniee H., On the Difference Between Consecutive Primes, Inventiones Math. 55 (1979), pp [4] Huxley M. N., On the Difference Between Consecutive Primes, Inventiones Math. 15 (1972), pp [5] Iwaniec H., Primes in Short Intervals, to appear. [6] Iwaniec H. an Jutila M., Primes in Short Intervals, ArJciv for Mat. ' 17 (1979), pp [7] Pintz J., On Primes in Short Intervals I, to appear. MAGDALEN COLLEGE, OXFORD OX1 4AU, ENGLAND