THE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS. IV. / = lim inf ^11-tl.

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1 THE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS. IV r. a. ran kin 1. Introduction. Let pn denote the nth prime, and let 0 be the lower bound of all positive numbers <r<> such that no Dirichlet /-.-function L(s, x) has a zero at s = <r-\-it, where <r><ro. Write / = lim inf ^11-tl.»->» log pn The purpose of this paper is to combine the methods earlier papers1 in order to prove the following theorem. Theorem. (1) / = c(l + 40)/5, where c< 42/43. used in two The precise definition of c is given by (18) and (8) below. The theorem is an improvement on the result /^(14-40)/5 obtained in II. It was shown in III that (2) I = d = < 57/59. The numbers c and d are connected by the relation 1 - d = (3 4-2"2)(1 - c)/4 > 1 - c, so that (1) is an improvement on (2) only if is not too close to unity, in fact, if 0< In particular, if the "grand Riemann hypothesis" is true, that is, if 0 = 1/2, we have (using the value of c given by (18)) / ^ 3c/5 < 109/186 = Notation. Since (2) is sharper than (1) for 0 = 1, we shall assume that 0<1, and write (3) a = (1 + 4 )/5. Let N be a large positive integer, and define (4) X = A^-'Oog Af)-»«+T) Received by the editors January 4, See [2] and [3]; I refer to these as II and III respectively. Numbers in brackets refer to the references cited at the end of the paper. 143

2 144 R. A. RANKIN [April as done in equation (6.7) of II; here 7 is a positive constant which may be taken to be 2. Let ö)(y) be the number of prime pairs p, p' which satisfy N < p' g p S 2N, p - p' = v, where v is an integer. We suppose that h, k, m, n and r are integers such that, for some positive A\, A2 independent of N, (5) Ai log TV < 2k/3 < k < k < A2\og N, (6) 0 iz m < n < A2 log N, r = k - h. The letters a, c, a, ß denote positive numbers independent of 7Y, and we write (7) D= 11Y1 + - Y and take t>i\ P(P-2)J' (8) X = > e-^aoa where 7 is Euler's constant and Ao is Buchstab's number (see relation (4) of III). For ix = 1, 2,, k we define a? as follows: (9) and write a ß (0 < u g r, k - r < n ^ k), if < n :g k - r), (10) *(0) = Ore*"1*. Then (11) I *(*) s = *(0) + 21) S(") cos ArvB, where 3. Lemmas. Lemma 1. We have

3 i95o] THE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS 145 f 2(r - v)a2 + 2vaß + (k 2r v)ß2 (0 = v < r), 00«2raß + (k 2r v)ß2 (r ^ v < k - 2r), (2r - k 4- v)a2 + 2(k-r- p)aß (k - 2r t% v < k - r), (k - v)ot2 (k-r v<k), and Z «W = T {2ra + (k - 2r)ß}2 + 0(log N). This follows easily from (9) and (11), and we omit the proof. It is in this lemma that the restriction k r = h>2k/3 made in (5) is used. Lemma l&qsx <p2(q) /-0.(/,8)-l II (^-4) + 0(.Z-i(log log X)2). Here ju(g) is the Möbius function, 6(q) is Euler's function, and D is defined by (7). It follows from the well known relation that m2(?) Z e*""" = Z * (4) /-<),(/.a)-i rf C2r.a) \a/ 8-1 <p2(?) /=0, (/,<,)-! = z m(?) n si </>2(?) p (2k,8) t^r n <«-» 8-1 <p W j>i (2».8) m(<?) - z n a-#). 9>X 4>2{q) p\{2v,q) By a straightforward reduction the first term on the right may be shown to equal - n (' ) D pi,.,>2 \p 2/

4 146 R. A. RANKIN [April See, for example, of [l]. Since l/<p(n) =0(w_1 log log n), the second term on the right is o/x-hlog log X)2 II (P- 1)1 = 0{xX-i(log log X)2}. V p\2v ' Lemma 3. 7/ s = 0 and *j independent of N, then "* II (^4) = > " + 0{ (log TV)'(log AO*}. The proof is similar to that of the lemma in III. We suppose that d is any product «,/?,, #>,-, of distinct odd primes, and let d' be the associated product We permit clearly, d' = (Pi, - 2)(pit - 2) (p^ - 2). d to take the value unity and then put d' = \. Then, Z^7=Z^7 + o{- (log log 3»)*1 d<n dd an dd \n ) = D + 0^- (log log 3n)*j, since 1/d' = 0{d~1 (log log d)2} for large'd; and n (r4) = n (.+7^ J>»,p>2 \P ~ 2/ J>».p>2 \ P 1/ For a given d', p is a multiple of <2, say v=iid, and, since m ^p< n, it takes all integer values in the interval m/d it<n/d, which we denote by 1(d). Clearly and tf+i _ m>+i (vd)-= - +0(n>), ngl(d) (s + l)d n-l «+! _ '-TT + (M')- _m * It is possible to dispense with the factor (log log 3»)1.

5 i95o] the difference between consecutive prime numbers 147 Hence 2 + n - r 7 x wor=m p\r,p>2 VP */ d<n «>igl(d).+1 _ 1 / _ 1 \ =-X 7+ o(»'2) ) d<n dä" \ d<n d' / M.+l _ m.+l ( /1 \) = 7+i r +0\~n~(log log 3w)// 4-0{»'(log log 3»)! log 2n} = D + 0{»'(log 2»)*}, and the lemma follows from (6). Lemma 4. fc-i = 2 fw + 0{log /Vdog log iv)1} = {2ra4- O - 2r)ß}s 4-0{log 7Y(log log AO2}. By Lemma 2, the left>hand member is equal to ~2lM II + o{*x-i(iog log z)2e (")} r-1 P\y.p>2 \P 2/ l y-l ) It follows from Lemmas 1 and 3 (with 5 = 0, 1) that this equals 2E W + 0{log 7Y(log log AO2} 4-0{X-lQog N)\log log X)2}, y~ 1 and the lemma follows from (4) and the second part of Lemma Proof of the theorem. By considering the expression where S(ß) = E log p (P prime), JV<pSJAT

6 148 R. A. RANKIN [April and ^(0) is denned by (10), it is possible to show that, for large N, I(N, k) = S(0)AMog/V{l4-<>(l)} 4-2 {14-0(1)} log»tf X)*f>)fir>) >»(. +.(«) is«sx ^ <r(?) /-o.a.«)-il *(Af \q/\ (12) 4- o(7y log2 N) «0)tf{l+ «(!)} Z 1S SX + 2JV{l+ «(!)}«Z lsasx M2(g) <p(?) JU2(9) «S(?) Z Z (")e4t<'//9 4- o(7y log2 TV). The proof of this is similar in every respect to the proof of (4.1) and (6.8) of II, except that ^(i)) is defined by (10) and is not ZJ=ig2T<'* as m 11- The restriction k <log N made in II is unessential and can be replaced by k < A 2 log N. It follows from (3), (4), (12), Lemma 4, and Lemma 4 of3 II that *-i 2{l4-o(l)} log2 N Z M«(2«0 i l > 47Y{14- o(l)} ZlW - 'N log #{H- *(1)}«(0) (13) 4- o(ar log2 N) = 27Y{2m4- (A - 2r)ß}2{l + o(l)} - <r/v log N{2rai + (k 2r)/32} {l 4" o(l)} 4- o(7y log2 N). Now, just as in III, it can be shown by Buchstab's method4 that, for sufficiently large N, form of Brun's * There is a misprint in the enunciation of the lemma which should read: "If X is large n*(q)/<p(q)~log X." 4 We must consider the interval (N, 2N) instead of the interval (x/log x, x). Our only reason for considering an interval of the form (N, 2N) is that it enables us to quote results directly from II. As far as Brun's method is concerned, the inequality stated remains true if we replace the left-hand side ü(2v) by the number of prime pairs p, p' which satisfy AN<p' p (A+l)N, p-p'~2v>0 where A^O.

7 i9jo] THE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS 149 ü(2v) < 4XAT Z?(log 7V)S l'.p>s n (f4> \p where X is defined by (8). It follows from Lemmas 1 and 3 that *=i 4X7Y *=i IN ) (14) e fm*(2r) < t e W + 0\-- (log log N)'\ r-k-r (log N)2 r-k-r Uog A" ) 2\a2r2N ( A7 ) + {ioi^(iogl0gm (log N) since r = 0(log N). Accordingly, if we make the assumption that «(2j»)=0 for all v<h=k r we have, from (13) and (14), that (2ra + (k 2r)ß) 2 { g* }11 + 0(1)1 1 (2ra2 +(k- 2r)ß2). <2x'-"(^)' 1 + '(", We now take a = l, ß = 2l,t, and put (16) KH1+ir)1,,g4 *-[{-('-t)hwhere a is a fixed positive number satisfying (17) a < 3-23'2. It is easily verified that, with this choice of a, h>2k/3. Then, for large N, aa r = - (2"2 4- Dllog A- 4-0(1), ZA 2ra 4- (k - 2r)ß = (2-21/2)( 4" 21'2A) = log N 4-0(1), 2ra2 4- (* - 2r)/32 = 2h = <r ^1 - log JV + 0(1).

8 150 r. a. rankin It follows from (15) that that is, that 1 1 / a\ l<ra(2^ + 1)) 2 <rj-<r2(l-j = 2X^ --\, 2 2 \ X/ \ 2X / a = 3-23's, which contradicts (17). We conclude, accordingly, that w(2v)>0 for some integer v satisfying It follows from this inequality, where and from (17), that / g <rc 42 (18) c = 1 - (3-23'2)/X = <, 43 and this proves the theorem. Note added 24 April Professor Erdös has informed me that considerable improvements have recently been made in Brun's method by A. Selberg and others; this will presumably mean that Buchstab's constant Ao, and consequently my constants c and d (see equations (2) and (18)), can be replaced by smaller values. References 1. G. H. Hardy and J. E. Littlewood, Partitio Numerorum III: On the expression of a number as the sum of primes, Acta Math. vol. 44 (1923) pp R. A. Rankin, The difference between consecutive prime numbers, II, Proc. Cambridge Philos. Soc. vol. 36 (1940) pp , The difference between consecutive prime numbers, III, J. London Math. Soc. vol. 22 (1947) pp Clare College, Cambridge

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