JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 4, Number I, January 1991
OSCILLATION THEOREMS FOR PRIMES IN ARITHMETIC PROGRESSIONS AND FOR SIFTING FUNCTIONS
JOHN FRIEDLANDER, ANDREW GRANVILLE, ADOLF HILDEBRAND, AND HELMUT MAIER
CONTENTS I. Introduction 1. Primes in arithmetic progressions 2. The main theorems II. Oscillation theorems for sifting functions 3. The statements 4. Proof of Theorem B2 5. Proof of Proposition 3.1 6. Proof of Proposition 3.2 7. Proof of Theorem C: preliminaries 8. Proof of Theorem C: completion 9. Proof of Theorem Bl 10. Proof of Theorem B3 III. Oscillation theorems for primes 11. Proof of Theorem Al 12. Proof of Theorem A2 13. Proofs of Theorem A3 and Proposition 2.1
I. INTRODUCTION
1. PRIMES IN ARITHMETIC PROGRESSIONS
For x ~ 2 real, q a positive integer, and a an integer coprime to q, let 0 and ~ be defined by (1.1) O(x; q, a) = E logp = qJZq) (1 +~(x; q, a)), p~x p=amodq Received by the editors June 15, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 11N13; Secondary llN25, 11N35. The first two authors were partially supported by NSERC and the last two authors were partially supported by NSF.
© 1991 American Mathematical Society 0894-0347/91 $1.00 + $.25 per page
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License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 26 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER
where rp is Euler's function. The prime number theorem for arithmetic progres- sions is equivalent to the statement that .1o(x; q, a) = 0(1) holds, as x -+ 00, for fixed q and a. The function .10 thus provides a measure for the regularity of distribution of primes in arithmetic progressions. The main focus in the study of primes in arithmetic progressions has previ- ously been on estimating .1o(x; q, a) from above uniformly for all, or almost all, q :5 Q, with Q as large as possible. The two fundamental theorems in this connection (cf. [Da)) are the Siegel-Walfisz and the Bombieri-Vinogradov theorems. It follows from the former that the bound:
-A (1.2) 1.1o(x; q, a)1 « (logx)
holds uniformly for q :5 (logx)B , for any fixed positive constants A and B. The latter states, in one form, that
( 1.3) "L..J -(-) 1 max 1.1o(x; q, a)1 « (logx) -A holds for any A > 0 with Q = y'X(logx)-B, where B is a suitable constant depending on A. As a consequence, the estimate (1.2) is true for all but o(Q) moduli q :5 Q, provided that Q :5 y'X(logx)-c with a sufficiently large con- stant C. It is generally expected that these results are far from being the best possible. In fact, under the Generalized Riemann Hypothesis, (1.2) is known to hold for all q :5 y'X(logx)-B and all a coprime with q, if B is a sufficiently large constant. However, even this conditional result probably is not optimal, and stronger conjectures about the size of .1o(x; q, a) and the range of validity of estimates like (1.2) or (1.3) have been put forward. Perhaps the best known of these conjectures is the Elliott-Halberstam con- jecture [EH], according to which the Bombieri-Vinogradov estimate is valid for Q = x 1- e and possibly even for Q = x(logX)-B with a sufficiently large con- stant B; see [EH, p. 62] for the precise formulation. This conjecture was at least partially motivated by probabilistic considerations; in fact, Elliott [E1] had shown earlier that, in an appropriate sense, "almost all" primelike se- quences of positive integers satisfy a Bombieri-Vinogradov type estimate (1.3) with Q = x(logx)-B, B = B(A). A corresponding conjecture for the estimate of individual terms .1o(x; q, a) has been made by Montgomery [Mo]. It implies that 1/2-e ( 1.4) 1.1o(x; q, a)1 «e (qjx) logx holds uniformly for q :5 x, for any given e > O. A weaker form of (1.4) may License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 27 be given as ( 1.5) The latter estimate is only of interest for q ~ x 1- e , while (1.4) remains non- trivial for moduli as large as x divided by a fixed power of log x . The strongest forms of both of these conjectures were recently disproved by Friedlander and Granville [FG]. They showed that for any A > 0 there exist arbitrarily large values of x and integers q ~ x(logx)-A and (a, q) = 1, for which I.:\(x; q, a)1 » 1, thus disproving Montgomery's conjecture (1.4). Moreover, such lower bounds hold for most moduli q, so that (1.2) even fails when averaged over a suitable range for q. As a consequence, the Bombieri- Vinogradov type estimate (1.3) cannot be valid with Q as large as x divided by any fixed power of log x . The main purpose of this paper is to extend these results by showing that the mentioned conjectures already fail for moduli that are significantly smaller than x divided by any fixed power of log x . Specifically, we shall prove that (1.4) fails to hold for all moduli q as small as x exp(-(logx) 1/5-0) , most moduli q as small as xexp(-(logx)I/3-0), and many moduli q as small as x exp( - (log x) 1/2-0), for any fixed 0 > 0, if the parameter e in (1.4) is suffi- ciently small. We also show that, given any positive constant A, the estimate (1.3) is false for all sufficiently large x with Q =x exp{ -(1-e)A(log2 x)2/log3x}, where throughout, logk x denotes the k times iterated logarithm. We state our main results in three theorems, Theorems A1-A3, which are proven by rather different methods. The first method is closest in spirit to the argument of Friedlander and Granville [FG]. It has its roots in a paper of Maier [Ma], in which he showed that the asymptotic formula n(x+y)-n(x) '" y/ logx fails to hold when y = (logx)A ,for any fixed positive;". The second method is a modification of the first, while the third is new and quite different from the other two approaches. These ideas are briefly discussed following the statements. Common to all three methods is the fact that they ultimately depend on oscillation results for certain sifting functions, a typical example being the well- known function: License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 28 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER 2. THE MAIN THEOREMS We consider here only the error function Ll( x; q , a) defined in (1.1). It is perhaps more customary to deal with the functions Li, ~ defined by lix ~ n(x; q, a) = I: 1 = q1(q)(1 +Ll(x; q, a)) p5,x p=amodq and x - If/(x; q, a) = A(n) = q1(q) (1 + Ll(x; q, a)). n (2.1 ) I:plq p<\ogq (2.2) q(logq) l+e < x:::; qexp (( logq )5/11-e ), there exist numbers x± with x/2 < x± :::; 2x and integers a± coprime with q such that (2.3) Ll(x' a) < _ -(I+e)OI(X,y) (2.4) -,q, -- y , where y = x/q and J1 (x, y) = log(logy / log2 x)/ log(logx/logy). The right-hand sides of (2.3) and (2.4) are» 1 if y = (logxf(1) , and are »y-(I+e)a/(I-a) if y = exp((logxt) with 0 < (); :::; 1/3. Thus the theorem shows that Montgomery's conjecture (1.4) is incorrect for moduli q as small as xexp(_(logX)I/3-0) , for any given positive J. The lower bounds in (2.3) and (2.4) as well as the range (2.2) for x are essentially the limits of what can be achieved by our method of proof. In connection with the lower bound in (2.2) it should be pointed out that for x < (1 - e)q1(q) logq , large lower bounds for Ll follow trivially from the fact that there are more reduced classes modulo q than there are primes :::; x. As far as the upper bound is concerned, we note that the theorem is not of much interest in the range q < x exp( _(logX)I/3+e), since then J 1 (x, y) > 1/2 and the lower bounds in (2.3) and (2.4) become smaller than the bounds predicted License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 29 by conjectures like (1.4). Under the assumption of the Generalized Riemann Hypothesis, the exponent 01 (x ,y) can be reduced to log(logy /1og2 x)/ 10g2 x . Theorem Al sharpens and extends to a much larger range the correspond- ing result of Friedlander and Granville [FG], and represents the analogue for arithmetic progressions of a recent result of Hildebrand and Maier [HM] on irregularities in the distribution of primes in short intervals. This latter result asserts that bounds of the type (2.3) and (2.4) hold with the same function 01 for all sufficiently large x and 2 :::; y :::; exp(A(10gx)I/3), if ~(x±; q, a±) is replaced by ~(x±, y) for suitable x±, x:::; x± :::; 2x, where ~(x , y) =.!. I: logp - 1 y x and A is a positive constant. The proof of Theorem Al combines the methods of [FG] and [HM]. The basic idea, which is essentially due to Maier [Ma], is to consider the occurrence of primes in a rectangular array rP + qs, where r and s run independently through the integers in certain intervals. Here q is the modulus of the pro- gression in which we are interested. Thus, for each fixed r and varying s, we are considering the primes in a progression modulo q and trying to prove that this number is not always close to what we "expect." On the other hand, for each fixed s and varying r, we are considering the primes in an arithmetic progression modulo P. Here P is chosen much smaller than q so that the number of such primes can be accurately estimated. This estimate, which rests on the Linnik-Gallagher prime number theorem, then reduces the problem to counting the number of s satisfying (s, P) = 1. Now, if this number is more (or less) than the expected number, then there will be correspondingly more (or less) primes than expected in some progression modulo q. Thus one is led to study oscillations of sifting functions of the type #{s :::; y: (s, P) = I}. In the case where P is of the form P = TIp License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 30 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER distinct prime factors less than log q and so almost all integers satisfy (2.1) (see, e.g., [E2, Chapter 4]). Condition (2.1) could probably be relaxed to some extent, but in this ap- proach it cannot be completely removed without sacrificing the quality of the result. Our next theorem gives a result that is somewhat weaker than the esti- mates of Theorem AI, but has the advantage of being valid for all q. Theorem A2. Let e > O. There exist N(e) > 0 and % = qo(e) > 0 such that for any q > qo and any x with N(e) 1/3 (2.5) q(logq) d(' ) > 1 -(I+e)02(x ,y) (2.6) x+ ' q, a+ - -1-5 -y , og x d(X' a) < __1_ -(i+e)02(X ,y) (2.7) -,q, -- 1 5 Y , og x where y = x/q and C>2(X, y) = 3log(logy/log2x)/10g(10gxlogy). This result is obtained by taking P in the above argument to be a product of primes chosen from an appropriate short interval. One consequence of this is that we are able to exclude all prime factors of q even if there are many of these. Another is that the close proximity of all prime factors of P allows us to deduce the oscillation result by combinatorial means. The factor l/log5 x is significant only if logy « (10g2 X)2. If two-sided estimates are not required and only a lower bound for Idl is sought, then this factor may be replaced by logy /(10g2 X)2 ; see the remarks after the proof. As compared with Theorem AI, the exponent C>2 is somewhat worse than C>I ' being less than 1 only for y < exp(logl/5-e x). (Assuming the Generalized Riemann Hypothesis this too may be improved.) However, Theorem A2 has the advantage of being valid for all q and has the additional advantage that the required oscillation result, Theorem B2 (see §§3, 4), is much easier to prove. Next we turn to lower bounds for averages over d(x; q, a). As almost all q satisfy the hypothesis of Theorem AI, we immediately obtain from Theorem Al the following result: Corollary 2.1. For all sufficiently large x and 2x exp( _(logx)5/II-e) :::; Q :::; x/2, we have '" 1 , -(I+e)o (x y) (2.8) £- -- rr,tax max Id(X ; q, a)1 »y 1', q:5:.Q rp(q) x:5:.x (a,q)=1 where y = x/Q. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 31 Theorem A2 yields a similar, but slightly weaker result. Stronger results may be obtained by bounding from below averages of the type (2.9) L _(1)1L\(x; q, a)1 = ~ L IO(X; q, a) _ x( )1 q (2.11 ) x exp( -cVlogx) < Q :::; x(10gx)-(1+e) , there exist Q± with Q/2 < Q± :::; 2Q and integers a± such that (2.12) _l_L\(X' a) > _1_ -(1+e)o3(x ,y) rp ( q ) , q, + - 10g2 X Y , (2.13) " _l_L\(x· q a ) < __1_ y -(1+e)03(X,y) L..J rp ( q ) "- - log x ' q"'Q_ 2 (q,a_)=1 where y = x/Q and 03 (X ,y) = 10g(10gy/log2 x)110g2 X. We remark that 03 is smaller, and hence better, than the exponents 01 and 02 from Theorems Al and A2, and indeed is the same exponent achieved by the method of Theorem A 1 on the assumption of the Generalized Riemann Hy- pothesis. The result remains of interest in the entire range y< exp( (log x) 1/ 2-e) and, with more effort, could be somewhat sharpened so that the bound remained larger than y-1/2 in the range y < exp(cvlogx) . The factor 1/log2x in (2.12) and (2.13) is significant only for logy « 10g2 x 10g3 x. In fact, for this latter range it can also be shown that this factor may be dropped, if one uses the method that gives Theorem AI, provided that the sum E q"'Q,(q,a±)=1 is replaced by the sum Eq",Q max(a,q)=1 (respectively E q"'Qmin(a,q)=1) • License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 32 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Since l3(X ,y) = (logX)(I+O(I))A if logy rv A(log2 X)2 / log3 x (A > 0 fixed), we deduce from Theorem A3 the following result, which sets a limit on the range of Q for which Bombieri-Vinogradov type estimates for averages of the form (2.9) may hold. Corollary 2.2. Let e > 0, A > O. For every x > xo(e, A), there exists an integer a such that for all Q with 2 (2.14) Q ~. x exp( -( 1 - e)A(log2 x) / log3 x), we have (2.15) ~ le(x; q, a) - qJ~)1 ~ I q~2 (e(x; q, a) - qJ~)) I > x(logx)-A. (q ,a)=l (q ,a)=l The method of proof of Theorem A3 is rather different from that of Theorems AI, A2. It is based on the following simple idea. We consider primes p rv x, and for given a we wish to study p == a (mod q) on average over q. This leads to a double sum over p and q. If we write p - a = qr, then the double sum can be transformed to a double sum over p and r. Since our values for q are almost as large as x, the complementary factor r is quite small, and the transformed sum deals with primes in progressions where the modulus r is sufficiently small so that analytic methods can be successfully applied. The resulting estimates lead to a sum that, in the special case that a is of the form TIp License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 33 Proposition 2.1. Fix A> o. Thenfor 2 ~ Q ~ !x, 0 < lal < (logx)A, we have (2.16) I fa (O(X; q, a)- qJ7q)) I «A r(a)Qlog(x/Q)+x(logx)-A+Q10glal. (q,a)=1 Here the last term on the right may be deleted unless a is prime. Thus, when Q satisfies both (2.11) and (2.14), (2.12) and (2.13) cannot hold with numbers a± in the range 0 < la±1 < (logX)A . We conclude this section with some speculation as to what might be the right upper bound for .1.(x; q, a) and formulate the following: Hypothesis. Let 8 > o. For all q, a, and x with q ~ %(8), (a, q) = 1, x ~ q(logq)l+e , we have (2.17) 1.1.(x; q, a)1 «y-(I-e)6(x,y), where y = x/q and ~( ) = . log(logy/log2 ~ (2.18) u x, y mm (!2' 1 X)) -_ mm . (!2' U3 (x, Y )) . og2 x It is equally reasonable to expect that the upper bound given by (2.17) and (2.18) holds for the short interval analogue where .1.(x; q, a) is replaced by .1. (x , y) = y-l(O(X + y) - O(x) - y). These hypotheses carry the implication that the results we obtain on average over q (Theorem A3) and, if GRH be assumed, the results we obtain here for individual progressions and those given for short intervals in [HM], are essentially best possible. As already indicated, the proof of Theorem Aj (1 ~ j ~ 3) is dependent upon a Theorem Bj, which demonstrates an oscillatory behavior for a certain sifting function of the type ~n$y,(n,p)=1 f(n). Those theorems are stated and proven in Chapter II. The proofs of Theorems Bl and B3 depend on a fur- ther Theorem C whose proof occupies the greater part of this paper. The final Chapter III gives the derivation of the above-stated results. II. OSCILLATION THEOREMS FOR SIFTING FUNCTIONS 3. THE STATEMENTS In this section we state our principal results on oscillations of (J>(y, z) and certain related functions. These results are proven in §§4-10. We first introduce some notation. For real numbers y > 0 and z ~ 2 and positive integers k, we set (J>k(y, z) = #{n ~ y: pin, p t k ~ p ~ z} = y II (1 - .!.) (1 + rk(y, z)), p License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 34 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER and let (y, z) = <1>1 (y, Z), r(y, Z) = r1(y, z). We further define a weighted form r(P\y, z) of r by y dt r(p )(y, z) = j r(P)(t, z)-, y/2 t where r(P)(y, z) is defined by n 1 ) (P) ¢(n)=YC ITI (I-pI (l+r (y,z)), n:9 p (3.2) ~u=IOgU+IOg2U+OC:::) (u;?:3), (3.3) :u~u = ~ + 0 (UI~gu) (u;?: 3) (cf. [HT, Lemma 1]). We can now state the two oscillation theorems for the functions rk and r(P) that are required for the proof of Theorems Al and A3. Theorem Bl. Let e > 0 and (y, z) be in the range (R1/6+e) with Uo = 1 + e, and let u = log y / log z. There exist numbers z ± satisfying (3.4) min(lz Z1-2/(u+2)) < z < z 2 ' - ± - , such that for every positive integer k satisfying (3.5) L 1 :::; 210g2 Z, p$.zplk License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 35 and for some y ± (that may depend on k) satisfying (3.6) y(1 - Ijlogy):::; y±:::; y, there hold rk(y +, z+) ~ exp{ -u~u - c u}, (3.7) { 1 rk(y _, z_) :::; - exp{ -u~u - c1 u}, where c1 is a positive constant depending at most on 8. Theorem B3. Let 8> 0 and (y, z) be in (R1/ 6H ) with Uo = 1 +8. There exist z± satisfYing (3.4) such that for every prime p and for some y± (which may depend on p) satisfYing (3.6) the inequalities (3.7) hold with rk replaced by f(P) . In what follows we are primarily concerned with the condition u > Uo for a large unspecified uo. How this assumption is relaxed to u ~ 1 + 8 is indicated at the end of this section. Actually, this weaker condition is necessary for the proofs of Theorems Al and A3 in that logl+e in (2.3) and (2.12) would otherwise have to be replaced by 10gUOH . The above results are quite sharp. It can be shown that, in the range (Re) , rk(y, z) is bounded from above by O(exp( -u~u + O(u))) (cf. [HM, Lemma 2] for the case k = 1). Thus, the lower bound (3.7) is best possible aside from the value of the constant c1 • Moreover, the localization of z± cannot be much improved. Indeed, it is clear from Theorem C and Proposition 3.2 that the numbers u± = logy j log z± cannot be better localized than within an interval of length 1. On the other hand, the localization of y ± is probably not best possible. In fact, under the Riemann Hypothesis, (3.7) can be shown to hold in the range (R 1/ 2+e) for any 8 > 0 with y ± = y (cf. the remarks following Corollary 3.1). For our applications, however, this deficiency is inconsequential. We remark that, instead of keeping y (essentially) fixed and varying z in order to obtain extreme values of rk(y, z) or f(P)(y, z), we could have instead fixed z and varied y. Specifically, under the conditions of Theorem B 1, it can be shown that the estimate (3.7) holds with z± = z and suitable numbers y ± satisfying y :::; y ± :::; Y Z2. A similar estimate, but with a much weaker localization of y ± ' was obtained in [HM] for the case of r(y, z). This was achieved by bounding from below certain weighted averages over the function f(u) = If(zu, z)12. This method, while being relatively simple, is not capable of yielding results of the quality of Theorems B 1 or B3, which will be needed for the proofs of Theorems A 1 and A3. Therefore it is necessary to use a different and more complicated approach. The restriction (3.5) in Theorem Bl is responsible for the corresponding restriction (2.1) in Theorem A 1. In order to deal with moduli k containing many small prime factors, it is natural to consider the analogue to (y, z) obtained by counting all those n :::; y whose prime factors lie in a given interval. In general, this should be more difficult to estimate but, in the case that the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 36 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER interval is rather short, a fairly precise oscillation theorem may be obtained by combinatorial means. The following such result is applied to Theorem A2. Theorem B2. For z 2:: zo' h ::; !z, k 2:: 1, P the product of any k primes, all of which are in the interval (z - h , z], we have (3.8) (_l)i-lrp(y)~(_l)i-l ( L 1- rp~)y) 2:: iy(~)z-i n9 (n,P)=1 for every integer j with 1 ::; j ::; k I 5 and every real y with (z - h)J 2:: y 2:: 4jzi l(k-j+l). We deduce Theorems B1 and B3 from a more fundamental result, Theorem C, and two auxiliary propositions. Theorem C gives an asymptotic formula for a "smoothed" version of r(y, z) in terms of the generating Dirichlet series: F(s) = F(z; s) = IT (1 - ~) ((s). p v -Av2/2 d r(y, z; A) = JXJ7C /00 r(ye, z)e v, v 2n -00 where A is a positive real parameter. We then have: Theorem C. Let e > 0 be fixed. For (y, z) in the range (Re) and log u ::; A::; exp(g3/2-e), we have (3.11) r(y, z; A) = Re p(y, z) + 0 CPi~~:)I) , License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 37 where u and.? are defined by (3.1) and the implied constant depends at most on e. For the proof of this result we use the saddle point method. Its basic idea is to write r(y, z) as a Perron integral over F(S)yS-1 Is along a vertical line segment whose abscissa is Re so' The main contribution to the integral then comes from small neighborhoods of the points So and So that are saddle points of the function F(S)/-I, and integration over these neighborhoods produces the main term Rep(y, z) in (3.11). This approach is very similar to the one used by Hildebrand and Tenenbaum [HT] in the "dual" problem of estimating the function: I{I(Y, z) = #{n:5 y: pin => P:5 z}. In fact, the main term in (3.7) is formally almost identical to the main term in their asymptotic formula for I{I(Y, z) , except that different Dirichlet series are involved. There are two important aspects in which the estimation of r(y, z) differs from that of I{I(Y, z) however. The first is that, while in the case of I{I(Y, z) there is a single dominating saddle point on the real line, the function F(S)yS-1 does not have a real saddle point, but rather two saddle points So and So off the real line whose contributions to the integral are complex conjugates. (The function has other saddle points as well, but their contribution turns out to be negligible.) This explains the appearance of the real part function in (3.11). The second difference between the estimations of r(y, z) and I{I(Y, z) is that in the case of I{I(Y, z), the generating Dirichlet series can be bounded trivially on any vertical line by its value at the abscissa. This is not possible in the case of ·r(y, z) and, as a result, the estimation of the integrand in the Perron integral at large heights becomes a nontrivial problem that prevents us from obtaining a formula of the type (3.11) for r(y, z) if u is large, say a fixed power of z. In order to obtain a better convergence in the Perron integral, we have to work with the smoother functions r(y, z; A) rather than with r(y, z) itself. To complement Theorem C, we prove the following results, which give esti- mates for the implicitly defined quantities So and p(y, z) in the main term of (3.11). Here e is an arbitrary given positive number, and the implied constants depend at most on e. Proposition 3.1. If (y, z) is in the range (RB) , then the equation (3.9) has a unique solution So = so(y, z) satisfying (3.10), and we have (3.12) so(y, z) = 1 - ~ + i; (1 + ;u) + O(.?I~g2 u)' (3.13) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 38 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Proposition 3.2. Suppose (y, z) is in the range (Re). Then we have (3.14) p(y, z) = exp{ -u~u + O(u)} , 2 (3.15) p(ZU-V, z) = p(zu, z)exp {V(~u - in) + 0Co~u) + 0(Vu )} (0 $ v $ (1 - e)u), (3.16) p(zu, z) = p(zu, z) exp { o(~ (IE(Z, z)1 + Z;2z + ZB) ) } (z $ Z $ 2z), (3.17) p(ZU, z) = p(zu , z) exp { 0(uI2) } (Z ~ z), where E(z, z) = L logp - (z - z). Theorem B2 is proven in §4. Propositions 3.1 and 3.2 and Theorem Care proven in §§5-8. In §§9 and 10 we deduce Theorems Bl and B3 from Theorem C and the two propositions. We conclude this section by deriving two simple corollaries of Theorem C and then reducing Theorems Bland B3 to the range u;::: uo' where Uo is arbitrary. First we show that the estimates of Theorem C hold with r(y, z) in place of r(y, z; A) , provided u = log y I log z is not too large. Corollary 3.1. Let e > 0 and (y, z) be in the range (Re), and suppose that u $ 2 3/2- e . Then we have r(y, z) = Re p(y, z) + O(lp(y, z)11 log u). To deduce this result, we apply Theorem C with e replaced by el2 and A = exp(2(3-e)/2). Without loss of generality, we may assume that e < 1/2. Using the estimates: r(yeV, z) = r(y, z) + O((lvl + lly)2) (Ivl $ 1) and r(y' , z) = 0(2) (y' > 0), which easily follow from the definition of r(y, z), we get r(y, z; A) = r(y~v'X [II e-).v 2/2 dv + 0(2v'X [~I (IVI + ~ )e-).v2 /2 dv) + o (2 VI foo e-).v 2/2 dV)' The error terms here can be bounded by «2 + 2 +2e-)'/2« exp{_u1H/3}« p(y, z) v'X y logu ' License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 39 if we use the definition of A, the bound u ::=; ,2>3/2-e , and the estimates (3.14) and (3.2) for p(y, z) and C;u' Moreover, the integral in the main term equals ill e-).v 2/2 dv = V;; (1 + O(e -)./2)) = ~n (1 + O(~ ) ). Thus, r(y, z; A) = r(y , z) ( 1 + 0 Co~ u) ) + 0 CPi~~:) I) and the estimate of the corollary follows from (3.11). We note that under the Riemann hypothesis, A can be taken as large as exp(zl/2-e) in Theorem C, and the estimate of the corollary can be shown to hold in the range (R I / 2+e) for any e > O. By appealing to Corollary 3.1 instead of Theorem C in the proof of Theorems B 1 and B3, one can then show that, under the Riemann Hypothesis, one can take y± = Y in (3.7), as we remarked earlier. Our second corollary concerns the asymptotic behavior of the Buchstab func- tion w(u), defined as the continuous solution of the system w(u)=lju (1::=;u::=;2), (uw(u))' = w(u - 1) (u> 2). This function was introduced by Buchstab [Bu] who showed that (3.18) lim r(zu, z) = eY(w(u) - e-Y) z-+oo holds for every u > 1 , where y is Euler's constant. Buchstab [Bu] also obtained the estimate: w(u) = e-Y+ O(exp{-uc;u + O(u)}). In view of this relation it is natural to consider the function W(u) = w(u) - e-Y. Maier [Ma] showed that W(u) changes sign in every interval of length 1. Cheer and Goldston [CG] proved that W(u) has at most two sign changes in any interval of length 1. Furthermore they showed that max W(v) = max W(v), min W(v) = min W(v). v~u u~v~u+2 v~u u~v~u+2 Theorem C and Proposition 3.2 enable us to describe the behavior of W(u) more precisely as follows. Corollary 3.2. There exists a decreasing function Wo(u) satisfying (3.19) Wo(u) = exp{-uc;u + O(u)} (u 2: uo) and a function 0 (u) satisfying O(u + t) - O(u) = tn(l + O(ljlogu)) (u 2: Uo ' 0 ::=; t ::=; 1) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 40 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER such that, as u -+ 00 , W(u) = Wo(u){cosO(u) + O(ljlogu)}. This result clearly exhibits the oscillatory behavior of W(u). Aside from a "decay factor" UiQ(u) , the function W(u) behaves like a trigonometric function with period 2. An immediate consequence of this behavior and the estimate (3.19) is that max W(v) = exp{ -u~u + O(u)} u~v~u+2 and similarly, min W(v) = - exp{ -u~u + O(u)}. u~v~u+2 To prove the corollary, we note that, by Buchstab's result (3.18) and Theorem C, W(u) = e-Y lim r(zu, z) z--+oo = e-Y lim {Rep(zu, z) + O(lp(zu, z)ljlogu)} , z--+oo provided u is sufficiently large. Also, the estimates (3.17), (3.14), and (3.15) of Proposition 3.2 imply that the limit l(u) = lim p(zu, z) z--+oo exists for every sufficiently large u and satisfies II(u)1 = exp{ -u~u + O(u)} and argl(u + t) - argl(u) = tn(1 + O(ljlogu)), (u ~ uo' 0 ~ t ~ 1), where argl(u) denotes a continuous branch of the argument of l(u). Taking Wo(u) = e-YII(u)1 and O(u) = argl(u) then yields the asserted estimates. As already mentioned, the results of Theorems Bland B3 follow fairly easily in the range 1 + e ~ u ~ Uo for fixed e and uo' In this range the formula (3.18) of Buchstab still holds with r replaced by rk (subject to (3.5)). In the case u ~ 2, this is an immediate consequence of [FG, Lemma 5]; in the case 1 + e ~ u ~ 2, that proof needs only a slight modification. An easy variation of Buchstab's original arguments shows that (3.18) also holds with r replaced by r(P) • Theorems Bl and B3 then follow for 1 + e ~ u ~ Uo from these facts in conjunction with Maier's result that W(u) changes sign in every interval of length 1 [Ma, Lemma 4]. Note. We take this opportunity to correct an error in [FG]. Professor Warlimont has kindly pointed out to us an omission from the identity in the proof of Lemma 5 in [FG]: specifically that the contribution "'£~/y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 41 In order to complete the proof of Lemma 5 it suffices to establish, for x ;::: y ;::: Yo' the estimate ' 3 q xf (3.20) ~ 1 « xl log y + -(-)rp q 1-'ogx n_x< By first considering the contribution to the sum of those n :::; XI/2 , then those n divisible by some prime > if , and finally the remaining n, we find that (compare Exercise 01 of[HaT]) for (1 = II flogy, ",' 1/2 2f "," 1/2 u L.J 1 «x + v(q)xly + L.J (nix ) n~x xlf2 ",* nU « ~ ",* _1_ « _q_ ~ exp { '" (_1__ .!.) } ~ logx ~ n l - u rp(q) logx pL.J~y2J pl-u P (3.22) q x « rp(q) logx' Therefore (3.21) and (3.22) yield (3.20), since x-u/2 « flogy/logx, and Lemma 5 follows as stated. 4. PROOF OF THEOREM B2 For m a positive integer, let s(m)=~.!., v(m)=~1. pimP plm We require the following lemma: Lemma 4.1. For any positive integers m, j with m squarefree and s( m) :::; j 12, we have (4.1) ~ .!. < 2s(~) ~ .!.. dim d - ] dim d v(d)~j+1 v(d)=j Proof. We first show that, for every integer i ;::: 0, 1 j! i 1 (4.2) ~ (j:::; ( . .),s (m) ~ (j' dim ] + l • dim v(d)=j+i v(d)=j We observe that any divisor d of m with v(d) = j + i can be written in License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 42 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER the form d = fSPI .. 'Pi (where v(fS) = j and the Pf.l are prime) in eji)i! = (j + i)!1 j! distinct ways, if we distinguish amongst different orderings of the Pf.l' By multiplying out and collecting terms on the right side, we get (4.2). To prove (4.1) we note that the left side is obtained by summing over i the left side of (4.2). The application of (4.2) thus yields " 1 "1,, j! i L...J d:5 L...J d L...J ( . + i)!s (m) dim dim i~1 ) lI(d)~i+1 v(d)=i :5 2: ! 2: e(~))i, dim d i~1 } lI(d)=i and since s(m):5 j/2, this gives (4.1). Proof of Theorem B2. If diP with v(d) ~ j, then d > (z - h)i ~ y, so [yld] = O. Now, since rp(Y) = 2: 1- {O~)y = 2:1l(d)([~] -~), n9 diP (n,P)=1 we have 1 (4.3) y 1. Irp(y)+(-l)i 2: ~1:5Y 2: -+d diP diP diP lI(d)=i lI(d)~J+1 lI(d)~J-1 Since s(P) :5 n(z)/(z - h) :5 1/8 for z > zo' we may use Lemma 4.1 to estimate the first term on the right-hand side of (4.3). Since C~I) :5 H~) for i :5 k I 3, the second term is :5 ~(~) :5 ~ ~. 2z } 2:diP lI(d)=J Substituting these in (4.3) we complete the proof of Theorem B2. 5. PROOF OF PROPOSITION 3.1 We begin with a lemma, which is needed again in later sections. Lemma 5.1. Let e > 0 and k ~ 1. Uniformly for z ~ 2 and s = (J + it in the range (5.1 ) e :5 (J < 1 - 1I:? , It I :5 1, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 43 we have r(1-S).'l' eW ( 1-0') (5.2) tp(z;s)=- 11 w dw + O ~k +0(1), (5.3) tp'(z; s) = (1 + O(;k)) ;~: + 0(1), (5.4) tp" (z; s) = (1 + o(~)) zl-s(1 - (1 ~ s)~) + 0(1), ~ (1 - s) 1 1-0' III ( ( )) Z 2 (5.5) Itp (z; s) 1:$ 1 + 0 . (1 _ (J)~ 1 _ (J ~ , where ~ = log z and the implied constants depend at most on e and k. Proof. Differentiating tp(z; s) = 10g(TIp (5.6) '( ) _ '" logp " (s) tp s - L.,.. ~ + r(s)' p 1 )) I-s 1 = ( 1 + 0 ( ~k Z 1 _ ~ + O( 1) , using partial summation and the prime number theorem with logarithmic error term. Substituting these estimates into (5.6) yields (5.3). The estimates (5.4) and (5.5) can be proved in the same way. To obtain (5.2), we note that by (5.3), s tp(z; s) - tp z; 1 - ~1) = l tp , (z; s , ) ds , ( 1-1/.'l' 1 r(l-s).'l' , ( w ) =-~11 tp z;l- ~ dw (I-S).'l' eW =- -dw l1 w + O(~~+ll(I-S).'l' I~ IldW1) + 0(1) r(I-S).'l' eW ( 1-0') =- 11 w dw + O ~k +0(1). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 44 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Since 9'(Z; 1- ~) = ];log (1- pl_11/.'? ) +IOg,(l- ~) 1 = - L I-I/.'? + 10g2' + 0(1) p holds for e/2::; a ::; 1-3/2' and 0::; t ::; 2TC/2'. Using Lemma 5.1 with e/2 in place of e and k = 1 , we see that in this range d / . d /. 9'''(a+it) -d arg9' (a + It) = -d 1m log 9' (a + It) = Re /( . ) t t 9'a+lt = Re ( 1 + 0 (~ ) ) C~ s - 2' ) + 0 a _ 2') -_ (1 (_12' )) ((1_a)2+t 1- 2 . Since I-a 1 2' 0< <--<- (1 - a) 2 + t 2 - 1 - a - 3 for 0 < a < 1-3/2' , (5.7) follows if z (and hence 2') is sufficiently large, as we may assume. Since arg 9'/ (a + it) is a continuously differentiable function of (a, t) in the range under consideration, the function t(a) defined by (i) is continuously differentiable as well. To prove (ii), we note that the function I (a) = 9''' (a + it(a))(1 + it' (a)) is real-valued, continuous, and has no zeros in the interval [e/2, 1-3/2'], since by (5.4), 9''' (a + it(a)) =I 0 in this interval. Thus f(a) is either monotonically License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 45 decreasing or monotonically increasing, as a increases from e/2 to 1 - 3/£7 . By Lemma 5.1 and the definition of t(a), we have f(e/2) = rp' (e/2 + it(e/2)) = -Irp' (e/2 + it(e/2))1 l-e/2 ( ( 1 )) = 11-e/~-it(e/2)1 1+0 £7 ' which is < _zl-e£7 , if z is sufficiently large. Similarly, f( 1 - ; ) = 1(3/£7) ~~::;_ 3/£7)1 (1 + O(~ )) , which is of order 0(£7). Assertion (ii) now follows. It remains to prove estimates (3.12) and (3.13). Setting So = 1 - sl/£7 , Sl = a l + itl , we can restate (3.12) as (5.8) a l = eu + 0(_12_)' tl = -n(1 + } ) + 0(_12-), log u "u log u From (3.9) and (5.3) we get -u£7 = rp' (z; so) = (1 + 0(~2 )) :~I £7 + 0(1), so that ( 5.9) Since lesil = /'1 and ISII = lal + itll = al (1 + O(t~/a~)) = al (1 + O(I/a~)), we deduce from (5.9) that (11/ e a l xu, which implies a l »logu and, by another application of (5.9), ~I = u( 1 + 0(~2) + O(u1 ) + O(:~)) (5.10) = u( 1 + 0 CO;2 u) ). Now note that the function eX / x is monotonically increasing in x 2:: 1 and that, by the definition of eu ' 2 /.+0 (ue u + 1)(1 + c5 + 0(c5 )) (5.11) eu +J = eu(1+ c5 leu ) 2 = u(I + (1 - I/eu)c5 + 0(c5 ) + O(I/ueu)) (1c51::; 1). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 46 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Comparison of (5.10) and (5.11) shows that a, = ~u + ~ with ~ = O( 1 j1og2 u) , as asserted in (5.8). To prove the second estimate of (5.8), we take the argument on both sides of (5.9), keeping in mind that by (3.10), t, = -to.:? lies in the interval (-2n, 0). This yields the equation -n + 0 ( - 1) + 0 (1)- = t - arctan -t, = t -- t, + 0 ( -1 ) , CL72 u.:?' 0" 0' ".2 ..z; " v, which implies the desired estimate t, = +~) + 0(_12 ) = + ; ) + 0(-\ ). -n(l a, log u -n(l "'u log u Finally, to obtain the estimate (3.13) for the derivative of so' we differentiate both sides of the equation (p' (z; so) = -u.:? with respect to u. In this way, we get so that () u .:? £lUSO(Z ,z) = - "( . ) u rp z, So _ (1-so)2.:? (1 +o( 1)) - z'-SO((l - so).:? - 1) .:? 1 (1+0( 1 1 (1+0(_1 )) =rp' (z ; so) (1 - o'o).:? ))= __u.:? log u by the estimates of Lemma 5.1 and (3.12). This proves (3.13). 6. PROOF OF PROPOSITION 3.2 First we consider the behavior of p(y, z) when z is fixed and y = ZU varies. Setting u eYySo-'P(s) j(u) = logp(z ,z) = log ..fiiU720 So nu/2 (6.1 ) 1 eY = (so - l)u.:? + rp(so) -logso - 210gu + log In/2' we can rewrite the first two estimates of Proposition 3.2 as (6.2) j(u) = -u~u + O(u) and 2 (6.3) j(u - v) = j(u) + v(~u - in) + 0Co~u) + 0(Vu ) (0 ::; v ::; u(l - e)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 47 respectively, and it suffices to show that these latter two estimates hold for 1-8 Uo ~ U ~ z Taking the derivative with respect to u in (6.1) we obtain / (u) = sou..z;'07 + (so - 1)..z;07'" + rp (so)so - so/so - Ij2u, where for brevity we have set s~ = (%u)so(ZU, z). Since rp'(so) = -u£' by (3.9) and (so - I)£' = -C;u + in + O( 1/log u) , ;~ = u~so (1 + O(IO~U)) = OCo~u) by Proposition 3.1, it follows that in the range (Rs) (6.4) /(u) = -C;u + in + 0(1/10gu). This implies, for 0 ~ v ~ (1 - e)u, f(u-v)=f(u)- fu~v/(W)dW = f(u) + fu~v (C;w - in + O(IO~W) ) dw = f(u) + v(C;u - in) + 0(v2/u) + O(v / log u), and hence (6.3), after adjusting the constant Uo in (Rs) , if necessary. Here we have used the estimate C;w = C;u + O(lu - wl/u) (Iu - wi ~ (1 - a)u), which follows from the bound (3.3) for (djdu)C;u. To prove (6.2), we apply (6.4) again to obtain f(u) = f(u ) + f" / (w) dw o 1uo = f(u ) - f"(C;w + O(I))dw o 1uo = f(uo) - uC;u + O(u) ; then (6.2) follows, if we can show that f(uo) = 0(1). Since, by Proposition 3.1, (so - I)£' = -c;u + 0(1), we see from (6.1) that f(uo) = rp(so) + 0(1), where So = so(ZUO , z). Now, by Lemma 5.1 and Proposition 3.1, we have {(I-So)..'? eW (ZI-crO ) rp(so) = - 11 W dw + 0 £,2 + 0(1) = O(e(l-cro)..'?) = O(iUo) = 0(1). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 48 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Hence we obtain f( uo) = O( 1) as desired. It remains to prove estimates (3.16) and (3.17). To this end we consider p(ZU, z) as a function of z with u fixed, and define, in analogy to (6.1), el' (6.5) g(z) = log p(zu , z) = (so - l)u log z + Setting So = so(ZU , z) and So = so(,zu , z) , the asserted estimates can be restated as (6.6) Ig(z) - g(z)1 « iIE(z, z)1 + ;2 + z~-e (z ~ z ~ 2z), and (6.7) Ig(z) - g(z)1 « ul£1 (z ~ z), where ~ = (z - z)lz. We shall prove that (6.6) holds in the case (6.8) The relation (6.6) for £1-3 ~ ~ ~ 1 and the relation (6.7) can thus be deduced by splitting the interval (z, z] into subintervals of the form (zl' zl (1 +£1-5)] (plus possibly one interval of shorter length) and by applying (6.6) (with e replaced by e12) along with the prime number theorem in the form E(ZI(l +£1-5), zl)« zl£1-7 to these intervals. From (6.5) we obtain where Rl = u(so - so) log z + To prove (6.6), we show that each of the terms R j is bounded by the right-hand side of (6.6). We first derive an estimate for So - so' By the definition of So and so' we have with '(~~) '( ~) '" logp R 4 = License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 49 Since « ZI-aO(A + Ze-I) « e~u (A + ze-I) « u(log U)(A + ze-I) by a standard upper bound for primes in short intervals and Proposition 3.1, it follows that IRsl « IR41 + ulog(z/z) (6.9) e-I « u(log U)(A + z ). On the other hand, since by Proposition 3.1 and our assumption (6.8), (6.10) so that IRsl = IlsO fIJI! (z; s) dsl » u.2"2 lso - sol. Comparing this bound with (6.9), we deduce that ~ log u ( (6.11) Iso-sol« .2"2 A+ zl-e1) • Since R3 = log(so/so) = O(lso - sol), R3 is bounded by the right-hand side of (6.6). To estimate R 1 , we write License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 50 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER and we note that by (3.9), the first term on the right-hand side is equal to u2'(.s0 - SO). It follows that by (6.11), (6.10), and (6.8). This proves the desired bound for RI . It remains to estimate R2 • We have R2=U(I-.s0)10g~- L 10g (l- ~o) z'5,p = u( 1 - .so) log ~ + . 1 "logp (6.12) z NSo 1 N L..J Z ogz z'5,p + L 1NLOgp N - ~o I) + O(lz - z + llz- 2uO ). o( z$.P The second error term here is of order « z2(I-uol (~ + ~) Z z2 2 2 u2 ( «(ulogu) (Az + Z21) «7 A+ z1) by Proposition 3.1 and the hypothesis u ~ zl-e , and the first can be bounded, using (6.8), by N-UO " 1 (N/) NI-uo z-z+l (N/) « z L..J og z z «z N 1og z Z z'5,p (1 ) ( AI) A Au log u AU log u Au « u og u L1 + z L1« 23 + z « 2 2 ' Moreover, since by Lemma 5.1and (3.9), NI-s zl-so = IZ : (I - .so) - So = (I + 0(1/22)),P'(z; .s0)(1 - .so) + 0(11 - .sol) = -(I + 0(1/22) + 0(I/u2))u(1 - .so)(logz) , License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 51 we have = -u(1 - so) (lOg ~ + E(Z; Z)) + 0(; ) + 0(Uil2) +0(,1 ~olil) +O(z~-e). Inserting these estimates into (6.12), we obtain u ~ uil 2 11 - So lil u IR21 « zIE(z, z)1 + 22 + uil + 2 + zl-e ' which proves the desired bound (6.6) for R2 in view of (6.8), and the fact that 11 - sol «eu/ 2 « 10gu/2 by Proposition 3.1 and (3.2). Thus the proof of Proposition 3.2 is complete. 7. PROOF OF THEOREM C: PRELIMINARIES In this section we derive estimates for the functions F(z; s) and rp(z; s) that are needed in the proof of Theorem C. We first state two auxiliary results from prime number theory. Lemma 7.1. Let 0 < e :::; 1 be fixed. Then we have, uniformly for z 2:: 2, 0:::; a < 1, and It I :::; exp((10gz)3/2-e), A() I-s 1 2: n~ =zl_~ +0 (I-a:_a exp(-(10gz)e/2) ) +0(1). n IC(a + it) I «max(l, Itl l - a ) 10g(1 + It I) , I,'(a + it) I «max(l, Itl l - a )10g2(1 + Itl). Proof. The first estimate is well known; see, e.g., [Iv, Theorem 1.9]. The second estimate follows from the first by Cauchy's integral formula. For the remainder of this section we let z 2:: 2 be fixed and write F(s) = F(z; s), rp(s) = rp(z; s), and 2 = log z. The next three lemmas give upper bounds for these functions for various ranges of values s. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 52 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Lemma 7.3. Let 0 < e :::; 1 be fixed. Then we have, uniformly for a ~ e and It I ~ 1, IF(s)l<< {2'lt ll /.2'IOg(1 + Itl) ifa ~ 1-1/2', l 2'ltll-O' log(1 + It!) exp(2z -0' /(1 - a)2') ife:::; a < 1 - 1/2'. Proof. Lemma 7.2 yields IF(s)1 = I((S) 11 (1 - ;s) I :::; 1((s)1 IT (1 + ~) p 1-0' « max(1, It I ) log(1 + It I) IT ( 1 + (f1 ) . p In the case a ~ 1 - 1/2' , the desired bound follows from the estimate In the remaining case e :::; a :::; 1 - 1/2' , we obtain the result from the estimates IT (1 + ~) « 1((a)(1 - a) IT (1 _ ~) I-I = exp( ~ ~e:(a)) p 1 )) r(I-O').2' eW IqJ(a)1 = ( 1 + 0 ( 2' JI W dw + 0(1) 1 )) 1-0' ( 1-0' ) :::; ( 1+0 ( 2' (l~a)2'+O (1~a)22'2 +0(1) 2z l -0' :::; (1 - a)2' + 0(1). Lemma 7.4. Let 0 < e :::; 1 be fixed. Umformly in the range (7.1) e :::; a :::; 1 - 1/2', 1 :::; It I :::; exp(2'3/2-e), we have (7.2) IF(s)1 « Itll-O' log2(1 + It I) exp((1 + e)zl-O' /2'), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 53 Proof. Setting ",(s) = Z)og (1 - ~) , p 1F'(s)1 = (1 -~) + C(s) (1 -~) 1?::11 I"(S) p 1 oo «ltI - loi(1 + It I) exp(I",(s)l) (1 + 1?;;?::1 /) by Lemma 7.2, (7.3) is seen to be a consequence of (7.4) and the bound ~ logp 1-00 (7.5) I L.....t -S-1 I «z . p = Z11-~ ~ 1 + O( ;~: exp( _2'8/2)) + 0(1). Since 11 - sl ~ It I ~ 1 and 1 - 0- ~ 1/2' by (7.1), this gives (7.5). The same estimate shows that for 0- :5 0-' :5 1 - 1/2' , we have 1""(0-' +it)1 = !?; p)~~~ 11 :5 (1 +O(~ ) )Zl-oo' +0(1). This implies that 1 k(o-+it)- ",(1- ~+it)!:5 (1 +O(~)) i zl-oo' do-' +0(1) :5 (1 + 0 (~ ) ) z;00 + O( 1) 1-00 Z :5 (1 + e) 2' + 0(1), and since I", (1 - ~ +it) I= I?; pl-l~2+it 1+ 0(1) = I~ pl+l~2+it 1+ 0(1) = 110g ,( 1 +~ +it) 1+ 0(1) :5 log2(1 +It I) +0(1) by Lemma 7.2, (7.4) follows. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 54 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER For the last lemma of this section we let e be a given positive number; y and z fixed numbers in the range (Re); and u = logy/log z. We further let So = so(y, z) = 0'0 + ito be defined as in the theorem by (3.9) and (3.10). The implied constants are allowed to depend at most on e. Lemma 7.5. For S = 0'0 + it, It I :::; 1, Iltl- tol > 1/2, we have 2 (7.6) IF(s)1 « IF (so) I exp( -u/310g u). Moreover, we have (7.7) IF(so)1 = exp{u(l + O(l/logu))}. Proof. Lemma 5.1 yields, for s = 0' + it as in the lemma, F(s) = exp(qJ(s)) = exp { - Ei((l - s)2) + o( -:;:) + 0(1)}, where V Ei(w) = fW e dv 11 v is a variant of the exponential integral. By Proposition 3.1 and (3.2) the first error term can be bounded by ZI-uO lou u log u U --4 «-4« --4- « -3-' 2 2 2 log u Thus, the estimates (7.6) and (7.7) are seen to be equivalent to (7.8) Re{Ei((1 - so)2) - Ei((1 - s)2)}:::; -u/310g2 u + 0(1) and (7.9) Re(Ei(1 - so)2) = -u(1 + 0(1/logu)), respectively, and it suffices to prove the last two estimates. We use the easily derived expansion (7.10) El(W). = e w(1- + 2'1 + -3 2 + 0 (1))4' W W w IRewl which holds uniformly in Rew ~ 1. Applying (7.10) with w = (1 - so)2 gives License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 55 Now, (7.12) by Lemma 5.1 and (3.9), and since I( 1 - so)2'1 x (1 - (Jo)2' x r;u x log u by Proposition 3.1, we see that (7.11) implies (7.9). A further application of Proposition 3.1 shows that and Re 1 = 1 (1 - so)22'2 (1 - (JO)2 £72 +0(_1_)=2..+10g3 U r;~ 0(_1_)10g3 U . Therefore, from (7.11) and (7.12) we obtain the estimate: (7.13) Re(Ei((1 - so)2')) = -U(1 + .; + 22 + 0(_13-)). "u r;u log u Next we apply (7.10) with w = (1 - (Jo - it)2' ,where It I :::; 1. Estimating each term on the right-hand side by its modulus in the same way as before, we obtain (7.14) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 56 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Thus the comparison of (7.13) with (7.14) shows that Re(Ei((I - so)£')) + I Ei((1 - s)£')1 ~ -~ + 2~u 0(+),log u and hence (7.8), holds in the range to + I/£' ~ It I ~ 1. Finally, to prove (7.8) for It I ~ to-I/£', we apply (7.10) with w = (1-s)£' to obtain Re(Ei((I - s)£')) = Re (1 zl-;£, + o( ZI-<1~ 2) - s (1 - 0 0 ) £' = (1 ~)::;£' cos(t£') + 0Co~u) = u( cos(t£') + 0Co~u))' Since -cos(t£')~O.9+0(I/logu) for It I ~ to-l/£'=(1l-I+O(I/logu))/£', this implies that - Re(Ei( (1 - s )£')) ~ I~ u ( 1 + 0 (10~ u ) ) , which, in conjunction with (7.13), proves (7.8) for the range It I ~ to - I/£'. 8. PROOF OF THEOREM C: COMPLETION We begin by expressing r(y, z) and r(y, z; 2) as Perron integrals over the functions F(s) = F(z; s). Lemma 8.1. For any fixed e > 0 and uniformly for y > 0, z;::: zo' e < 0 < 1 , and T;::: 1 , we have 1 l<1+iT F(s) s-) r(y, z)V(z) = -. Y ds+ 0 (10~ -) 2m <1-iT s T (8.1 ) lOgy) (( £' (2Z1-<1)) + 0 ( Y + 0 1 + Y1) Te/2 exp (1 _ o)£' ' where V(z) = IIp 1 f<1 l +iT F(s)yS ( 00 (y)<11 . ( 1)) -iT License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 57 Taking a1 = 1 + l/logy, the error term becomes y+ 1 ~ 1 ~ . ( y) « -r-L- nUl + L- mm 1, TI - nl n=1 y/2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 58 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Proof. Applying (8.1) with (J = (Jo' T = To, and yeV in place of y, we obtain r(y, z; A) V(z) = ~ roo r(yeV, z) V(z)e-Av2 /2 dv v 21t 1-00 1 10'0+iToF(s) s-I (3 ) = ~. Y k(s - 1) ds + L Ri ' 1tl O'o-ITo S ° i=1 where V1 00 V(S-I)-AV 2/2 d (S-I)2/2). k( s- 1) =.~ 1 e v=e v 21t -00 is defined as in the lemma by (8.3), RI = VI roo log(yev) e _AV 2 /2 dv 1-00 To . r;: 00 logy + eV -Av 2 /2 d logy «VA. 1 T. e v«3jB' -00 0 y R2 = VI roo log(y?) e _AV 2 /2 dv 1-00 ye «VIlogy roo (1 + eivi)e-AV2/2 dv « logy, y 1-00 y and V1 (2ZI-0'0) roo ( 1) -Av 2/2 R3 = T;/22 exp (1 _ (Jo)2 1-00 1 + yeV e dv 2 (2ZI-0'0) « l/2 exp (1 - (Jo)2 . The error terms RI and R2 are clearly of order O(logyjy) (assuming, as we may, that e ::; 2), and the same holds for the term R 3 , since 2 = log z « logy, and by Proposition 3.1, This proves (8.4). Lemma 8.3. We have 1 10'0+i F(s) s-I r(y,z;A)V(z)=-2' y k(s-l)ds 1tl O'o-i s (8.5) + O(yO'o-1 e(l-e/3; 1-0'0 (1og A)4 ). Proof. In view of the previous lemma, and since logy ys/2-1 yO'o-1 --«--<--y 2 - 2 ' License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 59 it suffices to show that the contribution of the range 1 ::; It I ::; To to the integral in (8.4) is bounded by the error term in (8.5). We split this range into the two parts 1 ::; It I ::; Tl and TI < It I ::; To, where TI = min(To' exp(2'3/2-e)). Since (0: - 1)2 - t2 ) /2/2). (8.6) Ik(O'o - 1 + it) I = exp ( 0 2A. «e- and (8.7) max IF(O'o + it) I « T~-O'qog(1 + To)2' exp ((12ZI-;~) It l:5 To - 0'0 « To2' exp(O(u)) «l/e by Lemma 7.3 and Proposition 3.1, the integral over the range TI < It I ::; To is bounded by « max IF(O'o + it)1 [To Ik(O'o - 1 + it)1 dt Itl:5 To iT! «l/e [00 e-t2/2Adt«l/eVie-T(/2A. iT! If A ::; To, then we have Tl ~ A by (8.2) and the definition of T1 , and the above bound becomes «l/eVie-Td2 «l/ee-z /2 « l/y, since TI ~ min(y, exp(2')) = z and z» (logy)1/(1-e/2) in the range (Re) . On the other hand, if A> To, then (8.2) implies exp(2'2/3-e) > To, in which case TI = To and the range TI < It I ::; To is empty. In either case, the contribution of this range is of the desired order, and to prove (8.5) it remains to show that the same is true for the range 1 ::; It I ::; Tl . Since F(s) = F(s) and k(s) = k(s), it suffices to consider the integral 0'0+iT! F(S)yS-1 1= 1 k(s-1)ds. O'o+i s Now 1 . 0'0- 11 = IY I' where with F(O' + it) (8.8) G(t) = o. k(O'o - 1 + it). 0'0 + It License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 60 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Hence, in order to prove the desired estimate, it suffices to show that (8.9) 1111 «£').1 1-0:o (1og).) 4 exp { ( 1 - '36)) u } . At this point a trivial bound of II by the integral over the modulus of G(t) would not be sufficient to yield (8.9). Therefore, we have to proceed more carefully, and we first apply an integration by parts to get yit 1 II = -'l-G(t) ITI - -'1-!TI o'(t)ylt. dt, lOgy I lOgy I which implies 1111« ~L!!}~~IIG(t)I+ lT1Io'(t)ldt}. We estimate the terms on the right-hand side using Lemmas 7.4 and 7.5. By Lemma 7.4 (with 6/3 in place of 6) and Proposition 3.1 we have, for 1~t~Ti' . 1-0: 2 { ( 6) Z 1-0'0 } IF(O'o + It)l<< t 010g (1 + t)exp 1 + '3 ~ « t 1-0:0 log 2 (1 + t) exp { ( 1 + '36) £'e'u ( 1 + 0 (1))log u } « tl-O'o 10l(1 + t) exp { (1 - i) u} , where in the last estimate we have used the identity e'u = u~u + 1 , the estimate (3.2) for ~u' and the bound logu ~ (1 - 6)£' . It follows that IF((1o + it)1 _t2/2)., max IG(t)l« max e t=I,T1 t=I,T1 t «exp{(1 - 6/3)u}. A similar argument shows that, for 1 ~ t ~ TI ' , . 1-0: 3 IF (0'0 + It)1 « t 0 log (1 + t) exp{(1 - 6/3)u} and hence, F '( . )1 _t2/2)., IF( ')1 _t2/2)., ( ) o'(t)1 I 0'0 + It e 0'0 + It e !!... l « t + t t+). -0: -flU 3 «t 0(1 + t/).)e log (1 + t) exp{(1 - 6/3)u}. It follows that 1 foo 2/2).,} 3 {( 6)} !I TI 10' (t)1 dt « {!).,dtI to'o + I 1)" te -t dt (1og).) exp 1 - '3 U 1-0: 4 «). o (log).) exp{(1 - 6/3)u}. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 61 Combining these estimates we obtain (8.9). Lemma 8.4. With c5 = 1I.:? we have 1 /to+o F(a + it)yGo-l+it r(y,z;A)V(z)=-Re o. k(ao -l+it)dt 'It to-o ao + It + O(yGo-l (I-B/3;I-GO(10gA)4) (8.10) e + o(yGO-~(so)1 exp ( - 410:2 u)). Proof. Since F(ao - it) = F(ao + it) and k(ao - 1 - it) = k(ao - 1 + it), the integrand in (8.10) changes into its complex conjugate when t is replaced by -t. Thus, the main term on the right-hand side of (8.10) can be written as 1 GO - i(to-O) Go +i(to+O) ) F(S)yS-1 -. (l + l k(s-l)ds. 2m Go-i(to+O) Go+i(to-o) S By Lemma 8.3, then, it suffices to show that the contribution of the ranges It I :::; to - c5 and to + c5 :::; It I :::; 1 to the integral in (8.5) is of the order of the second error term in (8.10). For the range It I :::; to - c5 the trivial estimate ro+i(to-O) F(S)yS-1 k(s - 1)1 «yGo-\ max IF(a it)1 1 o + } Go-i(to-o) S Itlsto-o is sufficient, since to« II.:? by Proposition 3.1, and max IF(ao+ it)l« !p(so)lexp(-uI310g2 u) ItlSto-o by Lemma 7.5. The second range to + c5 :::; It I :::; 1 requires a more delicate argument. It suffices to consider the integral 1 F( .) Go-I+it 1= / ao+lt~ k(ao -l+it)dt 10+0 ao + It = yGo-1 /1 G(t)/I dt = yGo-I II ' 10+0 say, where G(t) is defined by (8.8). We have to show that !P(so)1 ( (8.11 ) 1111« ~exp - u)2 • oZ 410g u Partial integration gives II « -11_ { max IG(t)1 + 1/1 G' (t)/I dtl}. ogy to+oStsl 10+0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 62 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER By (S.S), (S.6), and Lemma 7.5, we have (S.12) Thus, in order to prove (S.ll), it remains to show that the integral 1 , it 12 = / G (t)y dt 10+0 satisfies 2 (S.13) 1121 « IF(so)1 exp( -u/410g u). By (S.S), (S.6), Lemma 5.1, and Proposition 3.1, we have, for It I :::; 1, (S.14) 0'( ) = iF((Jo + it)k((Jo - 1 + it) {F'((Jo + it) __1_ k'((Jo - 1 + it)} t (Jo + it F((Jo + it) (Jo + it + k((Jo - 1 + it) = iG(t){¢'((Jo + it) + 0(1)} . {Zl-Uo-it (Zl-UO ) } = IG(t) 1 _ (Jo _ it + 0 (1 _ (Jo)2 + 0(1) -il = GI (t)z + O(uIG(t)I) , where iz l - uo GI(t) = 1- (Jo _ itG(t). In view of the bound (S.12), the contribution of the O-term to the integral 12 is of order 2 «u max IG(t)l« uIF(so)1 exp( -u/310g u), 10+0::;t~1 which is sufficient. Moreover, a further integration by parts yields (S.15) 1/1 G,(t)Z-il/1dtl«1 // ){ max IGI(t)I+/1 IG~(t)ldt}. 10+0 og Y z 10+0::;t~ I 10+0 Since log(y/z) = (u - 1)2» u2 if u 2: 2 , as we may assume, and since Zl-uo ulogu IGI (t)1 « 11 _ (Jo _ itIIG(t)1 « -ltl-IG(t)l, , Zl-uo Zl-uo , IGI (t)1 « . 2IG(t)1 + 11 . IIG (t)1 11 - (Jo - ltl - (Jo - It Zl-uo Zl-uo « -IG(t)1 + -(IGI(t)1 + ulG(t)l) tit 2 2 «u 102g uIG(t)1 t License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 63 by (8.14) and Proposition 3.1, the right-hand side of (8.15) is bounded by 1 2 2 «0;)1 max IG( t ) I{ --+u log U 1. u log2 U d t } ..z; to+t59$1 to to +15 t u2 log2 u « £' max IG(t)1 to to+t5 9$1 « u2(log u)2 IF (so)1 exp( -u/3log2 u), where the last estimate follows from (8.12) and the bound to » 1/£' (cf. (3.12)). This implies the desired bound (8.13) and completes the proof of Lemma 8.4. Lemma 8.5. With a = 1/£' we have to +15 F(a + it)yUo-l+it o . k(ao - 1 + it)dt 1.to-t5 ao + It (8.16) = (1 + 0(_1_)) ySo-IF(so)k(so - 1). log u so£' y'u/2n Proof. The integrand in (8.16) can be written as 1 )) ySo-1 k(s - 1) (8.17) ( 1+0 ( £' soo exp{i(t-to)logY+QJ(ao+it)}, since t = to+O(l/£') = 0(1/£') and k(ao-1+it) = k(so-1)(1+0(1/£')) in the range of integration. Expanding QJ(ao + it) in a Taylor series about t = to' we obtain i(t - to) logy + QJ(ao + it) = QJ(so) + i(t - to)(logy + QJ' (so)) - '21 ( t - to )2"()QJ So - 3ii ( -r - to )3 QJ III ( ao + l-r • ) with a suitable -r = -r(t), l-r - tol :$ It - tol. By the definition of so' the linear term here vanishes. Moreover, from Lemma 5.1 and Proposition 3.1, we see that " _Zl-So£, ( ( 1 )) QJ (so) = 1 _ So 1 + 0 (1 - ao)£' 2 = u£' (1 + O(ljlogu)) and o III • ZI-u £,2 ( ( IQJ (ao+l-r)I::; 1-ao 1+0 (i-a1o)£' )) =u£'3( 1+0 (1))logu . Thus we can write (8.17) as (8.18) (1 + O(l/£'))M exp{ -t(t - to)2u2'2 + R(t - to)}' where License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 64 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER and the function R(t) satisfies Setting 1 6 -6_ (IOgU)I/2--- _ (IOgU)I/2-- 1 U .!? u ' we have, for It - tol :::; 61' R(t - to) = 0(1), and exp(R(t - to)) = 1 + O(R(t - to)) = 1 + O((t - to)2.!?210;U) ' and for It - tol :::; 6 = II.!? , where the last estimate follows since 2/2 rOO r 1- 00 e- dr == v'2n and rOO e-r2/4 dr = /00 e-r2 /4 dr« e-(logu)/4 « _1_. 1t5l vu£' y'logu log u This proves (8.16). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 65 Proof of Theorem C. We now suppose that A satisfies the hypothesis log u :::; A:::; exp(2'3/2-e) of the theorem. Combining Lemmas 8.4 and 8.5 we obtain r(y z· A) _ Re /0-1F(so)k(so - 1) + o( yO"o-llF(so)1 ) , , - nV(z)so2' y'u/2n V(z)2' y'Ulogu yO"o-1 e(l-e/3)u A1-0"0 (IOgA)4) (8.19) ( + 0 V(z)2' ( yO"o-llF(so)1 ( u)) + 0 V(z)2' exp - 410g2 u . Since k(s - 1) = e(So-I//2}. = 1 + o(!) = 1 + 0(_1_) o A logu by the assumption A 2: log u of Theorem C, and since by Mertens' formula, the main term in (8.19) is equal to Re y~o-IF(So) +o(yO"o-IIF(So)l) =Rep(y, z)+o(IP(Y, Z)I). ne y soy'u/2n y'Ulog u log u The same argument shows that the first and third error terms are bounded by O(lp(y, z)l/ log u). Finally, using the seconds part of Lemma 7.5, the second error term is seen to be of order «yO"o-1 e(l-e/3)u A1-0"0(logA)4 = Y 0"0 -I IF (so) 1exp {e- 3u (1 + 0 (1)logu )} A1-0" °(lOgA) 4 « Ip(y, z)le -eu/4A1-0"0 (lOgA{ Altogether, we obtain r(y, z; A) = Re p(y, z) + o( 'Pi~~ :)1) + O(lp(y, z)le -eu/4AI-O"O(logA)\ This implies the desired estimate (3.11) in the case A satisfies, in addition to the hypotheses of the theorem, the inequalities A1-0"0 :::; eeu/8 and log A :::; u2 . Therefore it remains to show that the same estimate holds for values A in the range eu ) 1 1 c£l3/2-e (8.20) mm. (2u , 8(1 _ 0"0) :::; og/\.:::;..z; . Since 1 - 0"0 = (~u/2')(l + 0(1/log2 u)) «logu/2' by Lemma 5.1, the range (8.20) is only nonempty in the case u «2' , which we may henceforth assume. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 66 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER We shall show that (8.21) r(y, Z;A) =r(y, Z)+O()x) +o(~) holds for any A ~ 1. Now if A is in the range (8.20), then the error terms in (8.21) are bounded by 1 Z { eU} _u2/2 -(u-I)..2" v'J. + Y «exp - 16( 1 _ 0'0) + e + e 2 « exp { - ~~~ ( 1 + 0 Co~ u) ) }+ e_u /2 + e-(u-I)..2" « exp( _U3/2) « Ip(y, z)1 logu in view of Propositions 3.2 and 3.2, and the bound u«.!? Thus, setting Al = exp { min ( i, 8( /~ 0'0)) } , we obtain r(y z'A)=r(y Z'A )+o(IP(Y,Z)I) " " I logu = Rep(y z) + o(IP(Y, Z)I) , logu since (3.11) holds for A = AI' This shows that the desired estimate (3.11) remains valid in the range (8.20). To prove (8.21), we note that by sieve methods we have, for Y2 ~ YI > 0, cl>(Y2' z) - cl>(YI ' z) « (Y2 - YI + z) V(z). It follows that for Iv I :::; 1 , V v Icl>(ye ,z) cl>(y, z) I Ir(ye ,z) - r(y, z)1 = yeVV(z) - yV(z) cl>(yeV, z) v 1 v :::; yeV V(z) 11 - e 1+ yV(z) Icl>(ye ,z) - cl>(y, z)1 (yeV + z)lvl Yle v - 11 + z « v + '-'------'-- ye y z « Ivl +-.y Since for any y' > 0 , , cl>(y' , z) z Ir(y , z)1 :::; 1 + y'V(z) «1 + y' , License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 67 we obtain 9. PROOF OF THEOREM Bl We fix a (small) positive number e, real numbers y and z in the range Re , and a positive integer q. As before, we define u = logy I log z and £' = log z . Throughout this and the next section, the implied constants depend at most on e , and thus, in particular, are independent of k. Since the function rk and the hypothesis (3.5) of Theorem Bl depend only on the prime divisors of k that are < z , we may assume that k has no prime factors ~ z. Thus (3.5) reduces to (9.1) L 1 :5 2 log£'. plk We begin with the following simple lemma: Lemma 9.1. Let "f.,' denote summation over integers all of whose prime factors divide k. We have, for any a: ~ el2 and D ~ 1, , 1 £,1/2 L dO. « Do.(l-e/4) • d?D Proof. We have , 1 , 1 (d)o.(l-e/4) L d o. :5 L dO. D d?D d?1 = D-o.(l-e/4) ~'_1_ L- do.e/4 d?1 = D-o.(l-e/4) IT (1 __1_)-1 P Ol.e/4 plk License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 68 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Now if v is the number of prime factors of k and PII denotes the vth prime, then 1) -I ( 1) -I ( 1)-1 ( 2 g 1- pae /4 :::; p~I 1- pae /4 :::; XI 1- pe /8 (9.2) = exp {o( 2:: e~/8)} = exp{O(p~-e2/8)} p5,Pv p «exp(vl-e2/IO) «21/2, since p II « v log v and v « log 2 by (9.1). This proves the asserted estimate. Next, we define a smoothed form rk(y, z; A) of rk , analogous to the func- tion r(y, z ; A) in Theorem C, by 00 ..;x V -Av2/2 rk(y, z; A) = f'C 1 rk(ye, z)e dv. v2n -00 The following lemma relates the functions rk(y, z; A) and r(y, z; A). Here, and in the sequel, r.' has the same meaning as in Lemma 9.1. Lemma 9.2. For any A 2: 1 we have (9.3) rk(Y'Z;A)=rr(1-~){ 2::' ~r(~,z;A)+O( 1!4e/5)}' plk d5,yI-t/2 Y Proof. We start out with the identity 21/2) ( + 0 y' yl-3e/4 ' License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 69 where V(z) = I1p 'y' , 'Y') , ( -I ( , .;?1/2 ) 2: d = Y 2: '1(j + 0 ( 2: d = Y II 1 - Ii1) + 0 Y 1-38/4 . d'5,yl-./2 d~1 d>yl-./2 plk Y It follows that r ( , z) = II (1 -!) cl>k(Y' , z) - 1 k Y , plk P y'V(z) .;?1/2 ) } = II (1 - -1) { 2: '1(jr (y'd' z ) + 0 ( 1-38/4· plk P d-:;,yI-e/2 V(z)y Letting Y' = yeV and integrating both sides of this identity against the weight function J)..127C exp( _)..v2 12) , we obtain rk(y, z; )..) = II (1 -!) { 2:' ~r(~, z; )..) + o( .;?1:~38/4)}' plk P d which in view of the bound implies the asserted estimate (9.3). Lemma 9.3. We have and (9.5) where (9.6) Ck(y, z) = II (1-!) (1- I_~ /2)-1. plk P P u Proof. By the estimate (3.15) of Proposition 3.2 (with el2 in place of e), we have for d ::::; yl-e/2 , License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 70 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Hence the left-hand side of (9.4) equals , 1 (9.7) p(y, Z) L d l-C../2'+O(I/2') d$.yl-./2 ' 1 (' 1 ) = p(y, z) { L d l- eul2' + 0 L d l - e./2'+O(I/2') d~1 d~z 1 ~' logd )} + 0 ( 2' L..J 1-c../2' . d '1 ( 1)-I (1)-1 L d l - e./2' = II 1 - l-e./2' = ck(Y, z) II 1 - P d~1 plk P plk by the definition of ck(Y, z). Further, since 1 - ~u/2' ~ 2e/3 in the range Re (assuming that the constants Uo and Zo in the definition of R8 have been chosen sufficiently large), from Lemma 9.1 we get '1 '1 2'1/2 1 L dl-e./2'+O(I/2') «L d8/2 « z(8/2)(1-8/4) « 2'1/2 d~z d~z and 1 ~'logd 1 ~ logp~' 1 2' L..J dl-e./2' « 2' L..J p8/2 L..J d8/2 d ply, z)CklY, z)!J (1- ~) -I (1 + 0C2}1/')). This proves (9.4); the second estimate (9.5) can be established in the same way. For the remainder of this section we restrict (y, z) as in the statement of Theorem B1 to the range (R I /6+B)' This restriction would not be necessary if the Riemann Hypothesis were assumed. Lemma 9.4. There exist numbers z± in the range . ( 1-2/(u+2) I ) (9.8) mm z '2-z <- z ±-< z such that Re p(y, z+) ~ exp{ -u~u - ci u}, (9.9) { Re p(y, z_) ~ - exp{ -u~u - ci u} holds, where ci is a suitable positive constant depending at most on e. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 71 Proof. By Proposition 3.2 and (3.2) we have, for any z' in the range (9.8), Ip(y, z')1 = exp{ -u' ~UI + O(u'n, where u' = log y / log z' . Since u' = u log z, = u + 0 + logz (1 (.!u -1og1 z ) ) for min(z/2, ZI-2/(U+2)) :::; z' :::; z, we may replace u' by u in this estimate. To prove (9.9), it therefore suffices to show that there exist values z± satisfying (9.8) such that (9.10) 1Argp(y, z+)1 :::; n/3, 1Argp(y, z_) - nl :::; n/3, where Arg denotes the principal value of the argument. First we consider the case when u:::; U. By the estimate (3.15) of Propo- sition 3.2 we have, for u:::; u' :::; u + 2 , I argp(zu , z) = argp(zu, z) + n(u' - u) + O(I/10gu). Hence there exist numbers u± in [u, u + 2] such that (9.11) provided u 2': Uo with a sufficiently large constant uo' as we may assume. Setting z± = zu/u± , we then have Zu/(u+2) _< z± <_ z, so that (9.8) is satisfied, and argp(y, z±) = argp(z~± , z±) = argp(zu± , z) + O(u/2') by the last estimate of Proposition 3.2. In view of (9.11) and our assumption u:::; U, this implies (9.10). This argument would go through for u :::; Z5/12, if we would use Huxley's prime number theorem and the estimate (3.16) of Proposition 3.1, but it would break down for larger values of u. For the range U < u :::; z5/6-e , therefore, we use a somewhat different approach that depends on an "almost all" type estimate for primes in short intervals. Let h=zU/u, zn=z-nh (l:::;n:::;no =[z/2h]), so that, for 1 :::; n :::; no' z/2:::; zn :::; z, and zn - zn+1 = h » zl/6+e by our assumption that (y, z) lies in the range (R 1/ 6+e)' We shall show that (9.9) holds with z± = zn for suitable indices n± :::; no' ± First we note that the error term function E(z, z) defined in Proposition 3.2 satisfies z zh IE(t, t+h)ldt«A A lz/2 (log z) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 72 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER for any fixed constant A. This follows from Selberg's work [Se] and Huxley's zero-density estimate [Hu] in view of the bound h» zl/6+e; for the argument see Lemma 2 of Nicolas [Ni]. A simple consequence of this estimate is that 2 (9.12) IE(zn' zn+I)1 «hl2' holds for all n :5 no with at most 0(no/2'2) exceptions. If Z is large enough, this implies that there exist at least 2' consecutive indices n :5 no for which (9.12) holds, say n = n I + 1 , ... , n I + J , wher J = [2'] and 1 :5 nl :5 no - J . Let Uj = logyllogzn,+i' By the definition of zn and h we have, for j = 1, ... , J, logy logy Uj+1 - uj = log(zn _ (j + l)h) log(zn - jh) 1 1 u2'log((zn - jh)/(zn - (j + l)h)) hu 1 - 1 1 x-=-- - (logzn )2 + 0(2') z2' n' 1 Thus, in view of (3.15), the argument of p(z~j, Zn ) increases in steps of size 1 'I x lin, as j ranges over the integers in the interval [1, 2'}. It follows that, for suitable indices j± :5 J, we have (9.11) with zn in place of Z and 1 u± = uh . Using the esimate (9.12) for n = nl + j, 1:5 j :5 J, Proposition 3.2, and the definitions of hand J, we see that with z± = zn +j' = yl/U± , we 1 ± have argp(y, z±) argp(z~\j' ,zn +j' ) = 1 ± 1 ± nl+J ( h)) = argp(z~~, zn l ) + 0 ( ~ ~ IE(zn' zn+I)1 + 2'2 + ze n-nl = argp(zn~U , zn) + O\Jl( ZU ( 2'2he)) + Z = argp(z~~, Zn l ) + O( k ). This estimate, in conjunction with (9.11) (with zn in place of z), implies 1 (9.10), and thus completes the proof of Lemma 9.4. ProofofTheorem Bl. From Lemma 9.2, Theorem C, and Lemma 9.3 we obtain, for log U :5 A :5 exp(2'3/2-e) , rk(y, z; A) = ck(y, Z){ Rep(y, z) + 0CPi~~:)') + 0CP~;/:)I) } + 0(yll4E/5 ). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 73 The last error term here can be estimated, using the bound u ~ z5/6-e and Proposition 3.2, by Y -1+4e/5 « exp{ -( 1 - ~e )u2'} « exp{ -( 1 + leO )u~u} « Ip(y, z)1 «C (y z) Ip(y, z)1 logu k' logu' since ck(Y, z) ~ 1 . It follows that rk(y, z; J.) = ck(Y, Z){ Rep(y, z) + o('Pi~~:)') + o('P~;/:)I)}. We apply this estimate with YI = Y - y/2logy in place of y, and z± in place of z, where the numbers z± are chosen according to Lemma 9.4 so that (9.9) (with y replaced by y I) is satisfied. Adjusting the constant CI if necessary, we obtain (9.13) Note that the numbers z± here depend only on y and z, but not on k or J.. We now set ~ = 1/2logy, A = log(1 +~), so that a y~YI(1+~)=Yle , (9.14) y(1- lO~Y) ~ YI(1-~) ~ Yl e- a , and choose J. = (logy)4 = (U)-4. With this choice of parameters we have VI {a v -.l.v 2/2 (. f"7i = roo -.l.v 2/2 ) rk(YI' z+; J.) = v'2i J-a rk(YI e ,z+)e dv + 0 v A..z; Ja e dv VI fa v -Av 2/2 -Aa2 /2 = .I'C rk(Yl e , z+)e dv + O(.!£'e ) v27l' -a ~ max rk(Ylev , z+) + O(1/y) ivi::5a ~ max rk(y', z+) + O(1/y), y(I-ljlogy)::::;Y' ::5y and a similar estimate holds with z _ in place of z +. Combining these estimates with (9.14) yields max ±rk(y', z±) ~ exp{-u 10. PROOF OF THEOREM B3 Since the proof is to a large degree analogous to that of Theorem B 1, we keep the exposition short. We fix e > 0, real numbers y, z in the range Re , and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 74 1. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER a prime p ~ z. We concentrate on the case when p ~ z; the argument for p < z is similar and simpler. We write Cl>(p) ( z) = ",(P) ~ y, L...J rp(n) , n:;;x where E(P) denotes summation over integers n that are not divisible by p or by any prime < z , and c = C(P)(z) = p: (1 + p'(p,l_ 1))' p ;?::z p'>Jp We recall that r(P) and ,(P)(y, z) are defined by Cl>(P)(y, z) = yCV(P)(z)(l + r(P)(y, z)) and For A > 0 we set (P) vA 00 (p) v _;,v2/2 r (y, z; A) =..j2i 1-00 r (ye ,z)e dv, ,(P)(y, z; A) = r r(P)(t, z; A)~t. ly/2 The following lemma is analogous to Lemma 9.2. Lemma 10.1. For any A ~ 1 we have (10.1 ) (P) 1 { ",(p) ,i(d) (Y ) r (y, z; A) = C(1 _ 1/) L...J d (d) r d' z; A p d:;;yl-e/2 rp 1 ",(p) ,i(d) (Y .) } (.:?) - ~:;;7-e/2 drp(d) r dp' z, A + 0 yl-e/2 ' where the summation in E(P) is restricted to integers d that are not divisible by p or by any prime < z . Proof. Using the identity (n ~ 1), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 75 we obtain, for any y' > 0, (p) ( , ) - ~(P) ~ ,i(d) _ ~(p) f./(d) ~(p) 1 y, z - L...J L...J-(d) - ~ -(d) ~ n~y' din rp d~l rp n9'/d ~ ~t ~2(~i (~(~, z) -~(:») :F'~: ~(~i~ V(Z){ \- ~ + r(~, z) - H;~, z)} + O(i:/2) -Y '(p) { 1 ~(p) J.l2(d) (y' ) = Cy V (z) 1 + C(l _ 11) ~ d (d) r d' z P d9 1-,/2 rp 1 ~(p) J.l2(d) (y' ) } (y') - C(l - 1/p)p d~-'/2 drp(d/ dp' Z + 0 yl-B/2 ' where ~(p) J.l2(d) IT ( 1) C = ~ rp(d)d =, 1 + p'(p' _ 1) d~l p ~z p'~p is defined as above and the error term absorbs the contribution of the range d > yl-B/2 to the sum over fJ>(y'ld, z) and that of the range d > yl-B/2 to the sum over fJ>(y' I d p, z). It follows that (P) , 1 { ~(p) J.l2(d) (y' ) r (y, z) = C(1 _ lip) ~ drp(d) r d' z d~yl-'/2 1 ~(p) /(d) (y' )} (:f) - Pd~-'/2 drp(d) r dp' z + 0 yl-B/2 . Integrating both sides with y' = yeV against the weight function VA/2n' . exp( -AV 2 12) , we obtain (10.1). Lemma 10.2. We have and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 76 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Proof. This result can be proved in the same way as Lemma 9.2, using the estimates ~(P) Jj2(d) {( ~ 1 )} L..J d U ~(d) = exp 0 L..J p'u (p' _ 1) d?l p'?z = 1 + O(l/v'Z) (0' ~ e/2) and We omit the details. ProofofTheorem B3. From Lemma 10.1, Theorem C, Lemma 10.2, and Propo- sition 3.1, we deduce that, for logu :::; A:::; exp(2'3/2-e), r(P)(y, Z;A) = Rep(y, z)+O(lp(y, z)l/logu). It follows that y d ' y(P)(y, z; A) = r(p)(y', z; A)~ iy/2 Y y , dy' Y , dY') = Re p(y ,z)-, + 0 (1-1- i Ip(y ,z)I-, . iy/2 y ogu y/2 Y Since by Proposition 3.2, p(y', z) = p(y, z) (~) -eul.? (1 + oCo~u)) (y/2:::;y':::;y), we have eu Re r p(y', Z)d~' = 2 /.? -1 Rep(y, z) + o(IP(Y, Z)I). ly/2 y ~u/2' logu Replacing y by Yl = Y - y/2logy and applying Lemma 9.4, we deduce that ±y(P)(Yl' z±; A) ~ exp{ -u~u - c1 u} holds with suitable numbers z± satisfying (3.4) and an appropriate constant c1 • Taking A = (logy)4, we complete the proof of Theorem B3 by following the argument at the end of the preceding section. III. OSCILLATION THEOREMS FOR PRIMES 11. PROOF OF THEOREM Al For the proof of Theorem Al (as well as that of Theorem A2), we require the following result that refines Proposition 2 of [HM]. It is here that we appeal to the Linnik-Gallagher prime number theorem. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 77 Proposition 11.1. Fix e > o. For any squarefree integer n > 1. all of whose prime factors are ::; n I-e. there exists a divisor P of n. with niP prime. such that if (a, P) = 1. x ~ p2 • and x ~ h ~ x I exp( Vlog x). then (1.11) O(x + h ; P, a) - O(x; P, a) = ¢>~) {I + O(e -c1ogx/logP + e -c..jlogx)} , where c is a constant depending only on e. Proof. For a given value c > 0, a character X (mod q) is called" c-exceptional" if L(P, X) = 0 for some real P> l-cflogq. It is well known (cf. [Da, p. 94]) that there exists a constant Co > 0 for which there is at most one co-exceptional character X (mod q) for each q ~ 2. First we show that there is a divisor P of n, where niP is prime, for which there are no eco-exceptional characters X (mod P). Suppose that this is false and let ql be a divisor of n with nlql prime. Let Xl be the eco-exceptional character (mod ql) and let p be a prime divisor of the conductor of Xl. Set q2 = nip; note that p is a divisor of ql and so of n. Let X2 be the eco-exceptional character (mod q2)' and denote the characters of modulus n induced by Xl and X2 as X3 and X4' respectively. Since p divides the conductor of X3 but not that of X4' X3 and X4 cannot be the same character. However, since by our assumption L(Pi , Xi) = 0 for some numbers Pi > 1 - eco/logqi and i = 1, 2, we have L(Pi' Xi+2) = 0 where Pi> l-ecoflogqi ~ l-coflog n as each qi ~ ne • Thus X3 and X4 are distinct co-exceptional characters (mod n). This, as was already pointed out, cannot happen. Choosing P as above we have, under the hypotheses of the proposition, 1 O(x + h; P, a) - O(x; P, a) = (P) L x(a) L X(p) logp cp X (mod P) x · ) 0(· ) h h {-clogx/logp -c..jlogx} IO( x + h ,P, a - x, P, a - cp(P) I «cp(P) e + e . This proves (11. 1) . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 78 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Pro%/Theorem AI. We give the proof of (2.3), the proof of (2.4) being almost identical. Given q and x satisfying (2.1) and (2.2), we let Y = x/q and define z = log x/log Y so that (y, z) lies in the range (R 1/6+8) of Theorem B 1 (with Uo = 1 + e) provided that e is sufficiently small and %(e) sufficiently large. We next take z+ as in Theorem Bl, define n = TIp ( 11.4) R = 2~ exp( -y'logx). Let IMI denote the sum of the entries of L . For given s, the sum of entries in the sth column is O(2RP + qs; P, qs) - O(RP + qs; P, qs). This vanishes if (qs, P) > 1, and for those s with (qs, P) = 1 it may be estimated by applying (11.1) with x, h, and a replaced by P R, P R , and qs, respectively. By (11.2) and the definition of IMI = (~(Pp) (1 + O(e -clogRPflogP) + O(e -C"'IOgRP))k(Y + ' z+) (11.5) 'I' ~ Ry+ {I + expC-u License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 79 with a suitable constant c~ , since 10gRP x logx ~ logx = logy = ulogz 10gP z+ z and Vlog RP ~ Vlog x ~ log y = u log z by (11.4) and the definitions of z+ and z. On the other hand, the sum of the entries in row r is e(rP + qy+; q, rP) - e(rP; q, rP) = e(rP + qy+; q, rP), since rP:5 2RP < q by (11.4) and (2.2), and rP is not prime. This vanishes unless (r, q) = 1. Thus, (11.6) IMI = L e(rP + qy+; q, rP). r",R (r,q)=1 The number of r satisfying r",R and (r, q) = 1 isjust Rrp(q)/q+O(r(q)) , and, since R > XI/2 , the error is negligible. Thus, by (11.3), (11.5), and (11.6), there is at least one r for which . qy+ II e(rP + qy+, q, rP) ~ rp(q) {I + exp(-uc;u - c1 un, where C~' is an appropriate constant depending at most on e. Choosing this r and taking x+ = rP + qy+, we obtain d(X+; q, rP) ~ exp( -uC;u - C~' u) - 2rP /x+, which yields the desired estimate, since by (11.4) and (3.6), x+ = O(x exp( -Vlogx)) + qy(1 + O(ljlogy)) = x(1 + 0(1/ 10g2 x)), rP ~ - « exp( -y logx) « exp( -u log z) , x+ and by (3.2) and the definition 01 (x, y), uC;u = (1 + 0C:::) )UlOgU = (1 + o (10g2 U)) 10 log(logyjlogz) logu gy logz = (1 + 0C:::) )01 (x, y) logy. Remarks. Under the Generalized Riemann Hypothesis, (11.1) holds with the error replaced by 0(PXe- I / 2 ) and, in the above proof, we may choose z = (1/2)10gx leading to the conclusion of Theorem Al with 01(X, y) replaced by o (x ) = 10g(10gy / 10g2 x) 3 ,y 10g2 X License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 80 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER and the larger range q(logq)l+e:::; X:::; qexp((logq)1/2-e) in place of (2.2). As remarked in §2, we can use a modification of the above argument to delete the factor log2x from (2.12) and (2.13) of Theorem A3, at least for sums of the form E q ...... Q max(a, q)=l' E q ...... Q min(a, q)=l • Indeed, since we are able to assume logy « log2 x log3 x, the estimate of Corollary 3.1, in conjunction with (2.14), is sufficient for the proof. In this case we must choose y + above so that qy+ is independent of q and, for y this small, Corollary 3.1 allows us to do that. 12. PROOF OF THEOREM A2 Let y = xl q and u = log y /log(log x log y) so that 1/3 (12.1 ) N(e) « u :::; log x. Define v to be the positive real solution of (12.2) [(2 + 1~~2VX) 10gXr = y and let w = vllog2x. We pick j = [v] - 1 or j = [v] so that j is odd when we wish to prove (2.6) and even when we wish to prove (2.7). Clearly, if N(e) has been chosen sufficiently large, then Ij - ul < eul2 . 1/' Now take 1= y } /log x , h = 210gx, and z = (l + 2)logx, so that y = (z - h)} . As l/v 1 . 1 v ( v- . ) 1= -IYY /}- / = (2 + IOw)exp -._J logy , ogx JV we have (12.3) 2 + 1010: 2 x :::; I « w exp(4Iw) «logl/2 x, provided that N(e) is sufficiently large. Therefore, (12.4) /(1 + 21l)} :::; /e2jf/ :::; (logx)2/3(logx) 1/5 = (logX)13/15 as 2jll:::; tlog2x by the first inequality in (12.3). Let k = 1 + [clogxll0jlog2xlog(llog2x)] with c as in Proposition 11.1 with e = 1/2. Then, by the last inequality in (12.3), and by (12.4), k _ . + 1 > ~ log x _. J - 20 j(10g2 X)2 J > ~ (logX)2/15 j (1 + ~)} - 30 (log2 X)2 I ? 4jz) Iy so that } 4j} ( 12.5) (z-h) =y? (k_j+l)z. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 81 Let n be the product of any k + 1 primes in (z - h, z] that do not divide q. Note that, as Z5/8 ::; h ::; z, Huxley's theorem [Hu] gives n(z) - n(z - h) '" hi log z and so there are at least v(q) + k + 1 primes in (z - h , z]. Choose P as in Proposition 11.1, with e = 1/2. Define the matrix L exactly as in the proof of Theorem AI, except with y + replaced by y. By counting down each column, we see that the sum of the elements of L is L: RP {I + O(e -clogx/logP + e -c.jIOgXn n9 rp(P) (n,P)=1 by Proposition 11.1. Now c logx > clog x > 9 '10 (110 x) 10g2 x 10gP - klogz - J g g2 log(llogx) ~ 2j log(/ Ilog2 x) by (12.3), and cJlogx ~ 2j 10g(/1l0g2 x) . Therefore the sum of the elements of L is (12.6) R(y+ rp~/p(y)){1 +0(1/(/1l0g2x)2Jn. There are R¢(q) + O(r(q)) = Rrp(q) {I + 0(1 I (/llog X)2J n q q 2 row numbers r, with r '" Rand (q, rP) = 1; therefore we may choose some such row (say row ro) that contains more than (respectively less than) q ( P ) { I .2 2} rp(q) Y + rp(p)rp(Y) 1 + 0(1 (j 110g2 x ) n primes, in order to prove (2.6) (respectively (2.7)). Let Xo = x± = roP + qy and a = a± = roP' so that (a, q) = 1 . Now, as in the proof of Theorem AI, row ro contains exactly (j(xo; q, a) primes so that J-I. J-I P rp(Y) ( 1 ) (12.7) (-1) ~(xo,q,a)~(-I) (P)-+O ,2 2" rp Y (j 1l0g2 x) 1 By Theorem B2 and (12.5) we have that (_I/-lrp(y) ~ !(~)l» _1 (~k)J y 4 J zl v7 JZ » (/1l0g2 x ;~g(llOg2 x) r' where c' = ce120, and so, by (12.7), 2 (-1 /-1 ~(xo ; q, a) ~ 1I (U 110g2 X)(IH/3)u ( 12.8) ~ 1/(U3(IH/2)u 10g5 x) by the second inequality in (12.3). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 82 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Finally, as log u = {I + 0 ( I)} log(log Y /1og2 x) , we see that 3(1 + ej2)u log u :::; (1 + e)c52(x, y) logy, and so (2.6) and (2.7) both follow from (12.8). Remarks. In general this is the best possible result, using this method, up to the power of logx in the denominator. If logy:::; (1og2 X)2 then u :::; 10g2 x and so the factor exp(4jw) in the upper bound in (12.3) becomes relevant. It does not seem possible to remove this factor because of the I U in the denominator of the right-hand side of (12.8). More precisely, in order to use Theorem B2, we need I ~ 2 and so lil.Jo!; x >_ y >_ (/+2/logixk >1_ og i-I x. Now in the proof above we need two values of j, one odd and one even, so if y = logl x-I for some given integer J , we can take j = J and j = J - 1 only if I ~ (1ogX)I/(/-I). But then we are guaranteed the factor IU ::::::: (1ogX)I+o(l) in the denominator. If we only need to get a good estimate on one side (i.e. for 1.:1(x; q, a)l) when logy is small compared to (1og2 X)2 , then we do this with j = 1 + [log y j log( 2 log x)] and I = 2. Select h, k , and z as in the proof of Theorem A2, and then we can get the lower bound Ij(u2log2x)U(IH). If we were to suppose that the error term in Proposition 11.1 could be taken to B I 2 be O(PX - / ) (which would follow from the Generalized Riemann Hypothe- sis) then, in the proof of Theorem A2, we could take k = 1+ [log x j 1000 10g2 x] . This would imply that, in the definition of c52(x, y), we could replace the "3" by a "2" and (2.6) and (2.7) would still hold. As remarked in §2 we can also use Theorem B2 to prove a slightly weaker version of Theorem A3 and its Corollary. By just considering those integers q in [Q, 2Q] containing no prime factor in [2 logy , 410gy] and by taking 1= 1 in the proof of Theorem A2, one can, for example, deduce (2.15) in the range: 2 Q~xexp(-(I-e)B(1og2X) /1og3 X) , where A = 3B + 5 . 13. PROOFS OF THEOREM A3 AND PROPOSITION 2.1 We require the following simple result. Lemma 13.1. For any nonzero integer a and all x ~ 2, we have (13.1) I: ~~) = XCI (a) + O(r(a) log x) , m 1 -I ( 13.2) ~(m) = cl (a) logx + c2(a) + O(r(a)x logx) , m License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 83 where rp(a) c, (a) = -a- IT (1 + ( 1) _ 1) pfa P P and c2(a) « r(a). Proof. It suffices to prove (13.1) since (13.2) then follows by partial summation. Writing mjrp(m) = L.dlmIl2(d)jrp(d) and inverting the order of summation, we have 2 ~ = L 11 ~) L 1. m " 1 = ~ rp(a) + O(r(a)) ~ d a ' r'5.x/d (r,a)=' it follows that m rp(a) l(d) --=x-- rp(m) a L d (d) + O(r(a) logx). m for any c' < 2. Now let xexp(-cvlogx):S Q, :s xj2 and 0 < lal < Q,. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 84 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER Then L 8(x;q,a)= L L 10gp+O(Qlloglal) q,,-,Ql a '"'L..J 8(x; q, a) = '"'L..J rQ1 _ '"'L..J rQ1 q,,-,Ql r:5.x/2Q1 qJ(r) r"-'x/2Q1 qJ(r) (q,a)=1 (r,a)=1 (r,a)=1 + L qJ~r) + O(QIlog(lalx/QI))' r"-'x/2Q1 (r,a)=1 provided that (a, r l ) > 1 in the case that the exceptional modulus rl exists. Setting Sa(t) = L r/qJ(r) , r L 8(X;q,a)=x{ L qJ!q)+ L qJ(1q)d(X;q,a)} q,,-,Ql q,,-,Ql q,,-,Ql (q,a)=1 (q,a)=1 (q,a)=1 = x{ c1 (a) log 2 + L qJ!q)d(X; q, a)} + O(XI/2) q,,-,Ql (q,a)=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OSCILLATION THEOREMS 85 by Lemma 13.1 and the bound r(a)« lal'/3 :::; Q:/3 :::; X'/3, it follows that (13.4) L _l_Ll(x; q, a) = jX/QI (Sa(t) _ tc, (a)) d; + o(Q,IOg(la1x/Q,)). q~QI rp(q) x/2Q1 t x (q,a)=' Now, let x and Q be given subject to (2.11) and let y = x/Q. We give the proof of (2.13), the proof of (2.12) being almost identical. Set z = (1/2) logx . In case there is an exceptional modulus r, :::; exp( y'logx) ,let p be a prime divi- sor of r, ' otherwise let p = 2. By Theorem B3 there exist z_ in the range (3.4) and y_ in the range (3.6) so that ;:p(y_, z_) satisfies (3.7). Let Q_ = x/y_ and a_ = p I1p' X/Q_ dt j (Sa_ (t) - tc, (a_)) 2 = c, (a_);:P (y _ , z_). xj2Q_ t Thus, (2.13) follows from (13.4), (3.7), and the estimate c (a )>> rp(a_)>> II (1 _..!..) » _1_ » _1_. ,- a_ , pi log z _ log2 X p Proof of Proposition 2.1. In the case Q:::; X 2/ 3 , the result follows at once from Theorem 9 of [BFI] (for fixed a) or from Fouvry [Fo]; in fact, their results apply in a much larger range. In the case Q > x 2/ 3 , we follow the above argument except that instead of using the estimate (13.3) we bound 8(x; r, a) on average by means of the Bombieri-Vinogradov theorem. In this way, we obtain (13.4) with Q, = Q and an additional error term OA((logx)-A) on the right-hand side. Since by (13.1), r/Q(Sa(t) - tc,(a))d; = o( r/Q r(a)IOgt d;) = o(r(a) QIOg(X/Q)) , }Xj2Q t }X/2Q t x this proves the desired estimate (2.16). REFERENCES [BFIl E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), 203-251; II Math. Ann. 277 (1987),361-393; III J. Amer. Math. Soc. 2 (1989), 215-224. [Bul A. A. Buchstab, On an asymptotic estimate of the number of numbers of an arithmetic progression which are not divisible by relatively small prime numbers, Mat. Sb. (N.S.) 28 (1951),165-184. (Russian) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 86 J. FRIEDLANDER, A. GRANVILLE, A. HILDEBRAND, AND H. MAIER [CG) A. Y. Cheer and D. A. Goldston, A differential delay equation arising from the sieve of Eratosthenes, Math. Compo 55 (1990), 129-161. [Da) H. Davenport, Multiplicative number theory, 2nd ed., Springer.Verlag, New York, 1980. [El) P. D. T. A. Elliott, Some applications of a theorem of Raikov to number theory, J. Number Theory 2 (1970), 22-55. [E2) __ , Probabilistic number theory, Vol. I, Springer·Verlag, New York, 1979. [EH) P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Sympos. Math. IV Rome (1968-69), 59-72. [FG) J. Friedlander and A. Granville, Limitations to the equi·distribution ofprimes I, Ann. Math. 129 (1989), 363-382. [Fo) E. Fouvry, Sur Ie probleme des diviseurs de Titchmarch, J. Riene Angew. Math. 357 (1985), 51-76. [Ga) P. X. Gallagher, A large sieve density estimate near a = 1 , Invent. Math. 11 (1970), 329- 339. [HaT) R. R. Hall and G. Tenenbaum, Divisors, Cambridge Tracts in Math., vol. 90, 1988. [HM) A. Hildebrand and H. Maier, Irregularities in the distribution of primes in short intervals, J. Reine Angew. Math. 397 (1989), 162-193. [HT) A. Hildebrand and G. Tenenbaum, On integers free of large prime factors, Trans. Amer. Math. Soc. 296 (1986), 265-290. [Hu) M. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164-170. [Iv) A. Ivic, The Riemann zeta·junction, Wiley, New York, 1985. [Iw) H. Iwaniec, Rosser's sieve-bilinear forms of the remainder terms-some applications, Re· cent Progr. in Analytic Number Theory, Vol. I (H. Halberstam and C. Hooley, eds.), Aca· demic Press, London, 1982, pp. 203-230. [Mal H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), 221-225. [Mo) H. L. Montgomery, Problems concerning prime numbers, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, RI, 1976, pp. 307-310. [Ni) J. L. Nicolas, Repartition des nombres largements composes, Acta Arith. 34 (1979), 379-390. [Se) A. Selberg, On the normal density of primes in small intervals and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943),87-105. (J. Friedlander, A. Granville) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO, ONTARIO M5S lAl, CANADA DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS, URBANA, ILLINOIS 61801 DEPARTMENT OF MATHEMATICS, UNIVERSITY OF GEORGIA, ATHENS, GEORGIA 30602 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use O. Then for some c > 0, for all x> xo(e) , and all Q satisfying