arXiv:1606.08021v1 [math.NT] 26 Jun 2016 e al n ainedu e ´sla oruecas g´en´erale classe une r´esultat pour tel d’un variante ´etabli une ubro rm atr cutdwt utpiiy and multiplicity) with (counted factors prime of number .INTRODUCTION 1. Radziwi que Matom¨akil et exposer de travail vais S´eminaire au je maˆıtresses sous-jacentes ce de l`a de un discr´epance Dans d’Erd˝os. et, Enfin la conjecture de Liouville. ladite de de fonction logarithmique la version v´erifie une Tao de signes d de qu’`a celle comportements ainsi `a Chowla, veaux la ensuite de conjecture conduit la a moyennis´ees de Tao versions ult´erieure avec collaboration Leur tives. ´etait en croyance cette ´etabli que Radziwi ont l Matom¨aki et l ie`a valeur tive 8e eane,21-06 n 68`eme ann´ee, 2015-2016, S´eminaire BOURBAKI d ubro rm atr.Tu,i sacmltl utpiaiefun multiplicative completely a is it Thus, value factors. the prime of number odd R´esum´e rin onr` e re emeenmr evlus+ et +1 valeurs de mˆeme nombre pr`es le dive `a peu longueur donner de vraient intervalles les tous r´epandue, presque croyance etr ´ler eCol i u e valeurs les que dit c´el`ebre Chowla jecture de ruet translat´es par arguments rmes op´sae utpii´.O ’ted` eq’les com se qu’elle collection `a ce s’attend On multiplicit´e. compt´es avec premiers, function r iiil ytesur faprime. a of square the by divisible are h iuil function Liouville The .J eeceTkeadaortautl oec-esse fran¸cais en ci-dessus note fran¸cais la en traduit ci-dessus d’avoir Tokieda note r´esum remercie la ce Je traduit traduire d’avoir voulu 3. Tokieda bien remercie d’avoir Je Tokieda Prof. 2. gr´e au sais Je 1. : µ 1 2 (3) (2) (1) H IUIL UCINI HR INTERVALS SHORT IN FUNCTION LIOUVILLE THE − ≪ hc equals which , al´eatoire talprimes all at 1 λ ( n 1[resp. +1 = ) afnto eLiouville de fonction La ≫ esge,+ et +1 signes, de by λ atrMtmaiadRadziwi Matom¨akil and [after l] k λ o nsur-reitgr,adwiheul nitgr that integers on 0 equals which and integers, square-free on nir itnt xs n oreainnle eo n autre une Selon corr´elation nulle. ont fixes) distincts entiers annSOUNDARARAJAN Kannan sdfie ysetting by defined is p 1119 h iuil ucini lsl eae oteM¨obius the to related closely is function Liouville The . − ]si 1] n de nnmr ar[ep mar efacteurs de impair] [resp. pair nombre un admet − eat´qirbbe.Preepe n con- une exemple, ´etant Par ´equiprobables. 1 λ s n ocincompl`etement multiplica- fonction une est λ ( n λ et ) ( n if 1 = ) λ ( n )(lsg´en´eralement (plus en 1) + − . . l. ee fran¸cais.´e en n if 1 efntosmultiplica- fonctions de scmoe fa even an of composed is − fftvae td plus de et vraie, effet gatvr ’nn de- l’infini vers rgeant n de 1 ´su aconjecture r´esout la , ’xsec enou- de l’existence e qe-nsdsid´ees des lques-unes scmoe fan of composed is d´emonstration des ot om une comme porte λ ne rvi de travail rnier R´ecemment . to taking ction un2016 Juin 1119–02
The Liouville function takes the values 1 and 1 with about equal frequency: as − x →∞ (1) λ(n)= o(x), Xn x ≤ and this statement (or the closely related estimate n x µ(n)= o(x)) is equivalent to the prime number theorem. Much more is expectedP to≤ be true, and the sequence of values of λ(n) should appear more or less like a random sequence of 1. For example, ± one expects that the sum in (1) has “square-root cancelation”: for any ǫ> 0
1 +ǫ (2) λ(n)= O(x 2 ), Xn x ≤ and this bound is equivalent to the Riemann Hypothesis (for a more precise version of this equivalence see [32]). In particular, the Riemann Hypothesis implies that
1 +ǫ (3) λ(n)= o(h), provided h > x 2 , x
7 +ǫ (4) λ(n)= o(h), provided h > x 12 . x
(6) ψ(x)= Λ(n)= x + o(x), Xn x ≤ 1119–03 where Λ(n), the von Mangoldt function, equals log p if n > 1 is a power of the prime p, and 0 otherwise. Similarly, in analogy with (2), a classical equivalent formulation of the Riemann Hypothesis states that
1 2 (7) ψ(x)= x + O x 2 (log x) , so that a more precise version of (3) holds
(8) ψ(x + h) ψ(x)= Λ(n)= h + o(h), provided h > x1/2(log x)2+ǫ. − x 7 +ǫ (9) Λ(n) h, provided h > x 12 . ∼ x 2X 2 (10) Λ(n) h dx = o(Xh2), ZX − x Theorem 1.1 (Matom¨aki and Radziwil l[19]). — For any ǫ> 0 there exists H(ǫ) such that for all H(ǫ) < h X we have ≤ 2X 2 λ(n) dx ǫXh2. ZX ≤ x As mentioned earlier, the sequence λ(n) is expected to resemble a random 1 se- ± quence, and the expected square-root cancelation in the interval [1, x] and cancelation in short intervals [x, x + h] reflect the corresponding cancelations in random 1 se- ± quences. Another natural way to capture the apparent randomness of λ(n) is to fix a pattern of consecutive signs ǫ , ..., ǫ (each ǫ being 1) and ask for the number of n 1 k j ± such that λ(n + j)= ǫ for each 1 j k. If the Liouville function behaved randomly, j ≤ ≤ then one would expect that the density of n with this sign pattern should be 1/2k. Conjecture 1.2 (Chowla [2]). — Let k 1 be an integer, and let ǫ = 1 for ≥ j ± 1 j k. Then as N ≤ ≤ →∞ 1 (11) n N : λ(n + j)= ǫj for all 1 j k = + o(1) N. |{ ≤ ≤ ≤ }| 2k Moreover, if h , ..., h are any k distinct integers then, as N , 1 k →∞ (12) λ(n + h )λ(n + h ) λ(n + h )= o(N). 1 2 ··· k nXN ≤ k k Observe that j=1(1+ ǫjλ(n + j))=2 if λ(n + j)= ǫj, and 0 otherwise. Expanding this product out,Q and summing over n, it follows that (12) implies (11). It is also clear that (12) follows if (11) holds for all k. The Chowla conjectures resemble the Hardy-Littlewood conjectures on prime k- tuples, and little is known in their direction. The prime number theorem, in its equiv- alent form (1), shows that λ(n) = 1 and 1 about equally often, so that (11) holds for − k = 1. When k = 2, there are four possible patterns of signs for λ(n + 1) and λ(n + 2), and as a consequence of Theorem 1.1 it follows that each of these patterns appears a positive proportion of the time. For k = 3, Hildebrand [12] was able to show that all eight patterns of three consecutive signs occur infinitely often. By combining Hilde- brand’s ideas with the work in [19], Matom¨aki, Radziwil l,and Tao [22] have shown that all eight patterns appear a positive proportion of the time. It is still unknown whether all sixteen four term patterns of signs appear infinitely often (see [1] for some related work). Theorem 1.3 (Matom¨aki, Radziwil l,and Tao [22]). — For any of the eight choices of ǫ , ǫ , ǫ all 1 we have 1 2 3 ± 1 lim inf n N : λ(n + j)= ǫj, j =1, 2, 3 > 0. N →∞ N |{ ≤ }| We turn now to (12), which is currently open even in the simplest case of showing n N λ(n)λ(n +1) = o(N). By refining the ideas in [19], Matom¨aki, Radziwil l and PTao≤ [21] showed that a version of Chowla’s conjecture (12) holds if we permit a small averaging over the parameters h1, ..., hk. 1119–05 Theorem 1.4 (Matom¨aki, Radziwil l, and Tao [21]). — Let k be a natural number, and let ǫ> 0 be given. There exists h(ǫ, k) such that for all x h h(ǫ, k) we have ≥ ≥ k λ(n + h1) λ(n + hk) ǫh x. X X ··· ≤ 1 h1,...,hk h n x ≤ ≤ ≤ Building on the ideas in [19] and [22], and introducing further new ideas, Tao [34] has established a logarithmic version of Chowla’s conjecture (12) in the case k =2. A lovely and easily stated consequence of Tao’s work is λ(n)λ(n + 1) (13) = o(log x). n Xn x ≤ Results such as (13), together with their extensions to general multiplicative functions bounded by 1, form a crucial part of Tao’s remarkable resolution of the Erd˝os discrep- ancy problem [35]: If f is any function from the positive integers to 1, +1 then {− } n sup f(jd) = . d,n ∞ Xj=1 While we have so far confined ourselves to the Liouville function, the work of Matom¨aki and Radziwil l applies more broadly to general classes of multiplicative functions. For example, Theorem 1.1 holds in the following more general form. Let f be a multiplicative function with 1 f(n) 1 for all n. For any ǫ > 0 there exists − ≤ ≤ H(ǫ) such that if H(ǫ) < h X then ≤ h (14) f(n) f(n) ǫh, − X ≤ x Corollary 1.5 (Matom¨aki and Radziwil l [19]). — For every ǫ > 0, there exists a constant C(ǫ) such that for all large N, the interval [N, N + C(ǫ)√N] contains an integer all of whose prime factors are below N ǫ. Integers without large prime factors (called smooth or friable integers) have been extensively studied, and the existence of smooth numbers in short intervals is of interest in understanding the complexity of factoring algorithms. Previously Corollary 1.5 was only known conditionally on the Riemann hypothesis (see [33]). Further, (14) shows 1119–06 that almost all intervals with length tending to infinity contain the right density of smooth numbers (see Corollary 6 of [19]). Corollary 1.6 (Matom¨aki and Radziwil l[19]). — Let f be a real valued multiplicative function such that (i) f(p) < 0 for some prime p, and (ii) f(n) = 0 for a positive 6 proportion of integers n. Then for all large N the non-zero values of f(n) with n N ≤ exhibit a positive proportion of sign changes: precisely, for some δ > 0 and all large N, there are K δN integers 1 n < n <... Robert Lemke Oliver, Kaisa Matom¨aki, Maksym Radziwil l, Ho Chung Siu, and Frank Thorne for helpful remarks. 2. PRELIMINARIES 2.1. General Plancherel bounds Qualitatively there is no difference between the L2-estimate stated in Theorem 1.1 and the L1-estimate 2X λ(n) dx ǫXh. ZX ≤ x Lemma 2.1.— Let a(n) (for n =1, 2, 3 ...) denote a sequence of complex numbers and we suppose that a(n)=0 for large enough n. Define the associated Dirichlet polynomial A(y)= a(n)niy. Xn Let T 1 be a real number. Then ≥ ∞ 2 dx 2 ∞ sin(y/T ) 2 a(n) = A(y) 2 dy. Z0 x π Z | | y xe−1/TX f(x)= a(n), ex−1/T Xn ex+1/T ≤ ≤ so that its Fourier transform fˆ(ξ) is given by log n+1/T ∞ ixξ ixξ 2 sin(ξ/T ) fˆ(ξ)= f(x)e− dx = a(n) e− dx = A( ξ) . Z Zlog n 1/T − ξ −∞ Xn − The left side of the identity of the lemma is ∞ f(x) 2dx, and the right side is −∞ | | 1 ∞ ˆ 2 R 2π f(ξ) dξ, so that by Plancherel the stated identity holds. R−∞ | | In Lemma 2.1 we have considered the sequence a(n) in “multiplicatively” short in- 1/T 1/T tervals [xe− , xe ] which is best suited for applying Plancherel, whereas in Theorem 1.1 we are interested in “additively” short intervals [x, x + h]. A simple technical device (introduced by Saffari and Vaughan [30]) allows one to pass from the multiplicative situation to the additive one. 1119–08 Lemma 2.2.— Let X be large, and let a(n) and A(y) be as in Lemma 2.1, and suppose that a(n)=0 unless c X n c X for some positive constants c and c . Let h be a 1 ≤ ≤ 2 1 2 real number with 1 h c X/10. Then ≤ ≤ 1 2 2 2 ∞ c2 ∞ 2 h 1 a(n) dx X A(y) min 2 2 , 2 dy. Z0 ≪ c1 Z | | c1X y x Now in the inner integral over ν we substitute ν = δx, so that δ lies between h/(c2X) and 6h/(c1X). It follows that the quantity in (15) is c2X 6h/(c1X) (x(1 + δ)) (x) 2xdδ dx Z Z ≪ c1X/2 h/(c2X) |A −A | 6h/(c1X) c2X = (x(1 + δ)) (x) 2xdx dδ Z Z h/(c2X) c1X/2 |A −A | c2hX c2X dx 2 max (x(1 + δ)) (x) 2 , ≪ c h/(c2X) δ 6h/(c1X) Z |A −A | x 1 ≤ ≤ c1X/2 and now, appealing to Lemma 2.1 (with T = 2/ log(1 + δ) and noting that (sin(y/T )/y)2 min(1/T 2, 1/y2)), the stated result follows. ≪ 2.2. The Vinogradov-Korobov zero-free region The Vinogradov-Korobov zero-free region establishes that ζ(σ + it) = 0 in the region 6 2/3 1/3 σ 1 C(log(3 + t ))− (log log(3 + t ))− , ≥ − | | | | for a suitable positive constant C. Moreover, one can obtain good bounds for 1/ζ(s) in this region; see Theorem 8.29 of [15]. Lemma 2.3.— For any δ > 0, uniformly in t we have log x λ(n)nit x exp , 2 +δ ≪ − 3 Xn x (log(x + t )) ≤ | | and π(x) log x pit + x exp . 2 +δ ≪ 1+ t − 3 Xp x (log(x + t )) ≤ | | | | 1119–09 Proof. — Perron’s formula shows that, with c =1+1/ log x, 1 c+ix ζ(2w 2it) xw λ(n)nit = − dw + O(xǫ). 2πi Z ζ(w it) w Xn x c ix ≤ − − 2/3 δ Move the line of integration to Re(w)=1 (log(x+ t ))− − , staying within the zero- − | | free region for ζ(w it). Using the bounds in Theorem 8.29 of [15], the first statement − of the lemma follows. The second is similar. For reference, let us note that the Riemann hypothesis gives uniformly π(x) (16) pit + x1/2 log(x + t ). ≪ 1+ t | | Xp x ≤ | | 2.3. Mean values of Dirichlet polynomials Lemma 2.4.— For any complex numbers a(n) we have T 2 a(n)nit dt (T + N) a(n) 2. Z T X ≪ X | | − n N n N ≤ ≤ Proof. — This mean value theorem for Dirichlet polynomials can be readily derived from the Plancherel bound Lemma 2.1, or see Theorem 9.1 of [15]. We shall draw upon Lemma 2.4 many times; one important way in which it is useful is to bound the measure of the set on which a Dirichlet polynomial over the primes can be large. Lemma 2.5.— Let T be large, and 2 P T . Let a(p) be any sequence of complex ≤ ≤ numbers, defined on primes p, with a(p) 1. Let V 3 be a real number and let | | ≤ it≥ E denote the set of values t T such that p P a(p)p π(P )/V . Then | | ≤ | ≤ | ≥ P (V 2 log T )1+(log T )/(log P ). |E| ≪ Proof. — Let k = (log T )/(log P ) so that P k T . Write ⌈ ⌉ ≥ k it it a(p)p = ak(n)n . pXP nXP k ≤ ≤ Note that a (n) k! and that | k | ≤ k a (n) a(p) π(P )k. | k | ≤ | | ≤ nXP k pXP ≤ ≤ Therefore, using Lemma 2.4, we obtain 2k T 2k π(P ) it k 2 k k a(p)p dt (T + P ) ak(n) k!P π(P ) . |E| V ≤ Z T ≪ | | ≪ − pXP nXP k ≤ ≤ The lemma follows from the prime number theorem and Stirling’s formula. 1119–10 2.4. The Hal´asz-Montgomery bound The mean value theorem of Lemma 2.4 gives a satisfactory bound when averaging over all t T . We shall encounter averages of Dirichlet polynomials restricted to | | ≤ certain small exceptional sets of values t [ T, T ]. In such situations, an idea going ∈ − back to Hal´asz and Montgomery, developed in connection with zero-density results, is extremely useful (see Theorem 7.8 of [23], or Theorem 9.6 of [15]). Lemma 2.6.— Let T be large, and be a measurable subset of [ T, T ]. Then for any E − complex numbers a(n) 2 it 1 2 a(n)n dt (N + T 2 log T ) a(n) . Z X ≪ |E| X | | E n N n N ≤ ≤ it Proof. — Let I denote the integral to be estimated, and let A(t) = n N a(n)n . Then P ≤ it it I = a(n)n− A(t)dt a(n) A(t)n− dt . Z X ≤ X | | Z E n N n N E ≤ ≤ Using Cauchy-Schwarz we obtain 2 2 2 n it (17) I a(n) 2 A(t)n− dt , ≤ | | − N Z nXN nX2N E ≤ ≤ where we have taken advantage of positivity to smooth the sum over n in the second sum a little. Expanding out the integral, the second term in (17) is bounded by n i(t2 t1) (18) A(t )A(t ) 2 n − dt dt . Z 1 2 − N 1 2 t1,t2 nX2N ∈E ≤ Now a simple argument (akin to the P´olya-Vinogradov inequality) shows that n N (19) 2 nit +(1+ t )1/2 log(2 + t ); − N ≪ 1+ t 2 | | | | nX2N ≤ | | here the smoothing in n allows us to save 1+ t 2 in the first term, while the unsmoothed | | sum would have N/(1 + t ) instead (see the proof of Theorem 7.8 in [23]). Using this, | | and bounding A(t )A(t ) by A(t ) 2 + A(t ) 2, we see that the second term in (17) is | 1 2 | | 1 | | 2 | N A(t ) 2 + T 1/2 log T dt dt N + T 1/2 log T I. Z 1 Z 2 2 1 ≪ t1 | | t2 1+ t1 t2 ≪ |E| ∈E ∈E | − | Inserting this in (17), the lemma follows. 1119–11 3. A FIRST ATTACK ON THEOREM 1.1 In this section we establish Theorem 1.1 in the restricted range h exp((log X)3/4). ≥ This already includes the range h>Xǫ for any ǫ> 0, and moreover the proof is simple, depending only on Lemmas 2.2, 2.3 and 2.4. Since a large interval may be broken down into several smaller intervals, we may assume that h √X. ≤ Let denote the set of primes in the interval from exp((log h)9/10) to h. Let us P further partition the primes in into dyadic intervals. Thus, let denote the primes P Pj in lying between P = 2j exp((log h)9/10) and P = 2j+1 exp((log h)9/10). Here j P j j+1 runs from 0 to J = (log h (log h)9/10)/ log 2 . This choice of was made with ⌊ − ⌋ P two requirements in mind: all elements in are below h, and all are larger than P exp((log h)9/10) which is larger than exp((log X)27/40), and note that 27/40 is a little larger than 2/3 (anticipating an application of Lemma 2.3). Now define a sequence a(n) by setting (20) A(y)= a(n)niy = λ(p)piyλ(m)miy. Xn Xj pXj X/P Xm 2X/P ∈P j+1≤ ≤ j In other words, a(n) = 0 unless X/2 n 4X, and in the range X n 2X we have ≤ ≤ ≤ ≤ (21) a(n)= λ(n)ω (n), where ω (n)= 1. P P Xp p∈Pn | Tur´an’s proof of the Hardy-Ramanujan theorem can easily be adapted to show that for n [X, 2X] the quantity ω (n) is usually close to its average which is about ∈ P W ( )= p 1/p (1/10) log log h. Precisely, P ∈P ∼ P 2 (22) ω (n) W ( ) XW ( ) X log log h. P − P ≪ P ≪ X Xn 2X ≤ ≤ Moreover, note that for all X/2 n 4X one has a(n) ω (n) and so ≤ ≤ | | ≤ P (23) a(n) 2 XW ( )2. | | ≪ P Xn Now 2X 2 2X 2 W ( )2 λ(n) dx λ(n)ω (n) dx P Z ≪ Z P X x 2X h (ω (n) W ( ))2dx Xh2W ( ). ≪ Z P − P ≪ P X x Combining this with Lemma 2.2 we conclude that 2X 2 2 2 X ∞ h 1 Xh (24) λ(n) dx A(y) 2 min , dy + . Z ≪ W ( )2 Z | | X2 y2 W ( ) X x Thus, setting X 2 2 2 2 iy iy h 1 (26) Ij = (log Pj) p λ(m)m min 2 , 2 dy, Z X X X X y − p j X/Pj+1 m 2X/Pj ∈P ≤ ≤ and noting that 1/ log P W ( ), we obtain j j ≪ P P X 2 J 2 h 1 1 2 (27) A(y) min 2 , 2 dy W ( ) Ij W ( ) max Ij. Z X | | X y ≪ P log Pj ≪ P 0 j J − Xj=0 ≤ ≤ To estimate Ij, we now invoke Lemma 2.3. As noted earlier, our assumption that h exp((log X)3/4) gives log P (log h)9/10 (log X)27/40. Thus for y X, Lemma ≥ j ≥ ≥ | | ≤ 2.3 shows that iy Pj 1 27 2 δ Pj 1 1 p + Pj exp (log X) 40 − 3 − + , ≪ log Pj 1+ y − ≪ log Pj 1+ y log Pj pXj ∈P | | | | say. Using this bound for X y log P , we see that this portion of the integral ≥ | | ≥ j contributes to Ij an amount P 2 2 h2 1 j λ(m)miy min , dy. (log P )2 Z X2 y2 ≪ j log Pj y X X ≤| |≤ X/Pj+1 m 2X/Pj ≤ ≤ Split the integral into ranges y X/h, and 2kX/h y 2k+1X/h (for k = 0, ..., | | ≤ ≤ | | ≤ (log h)/ log 2 ) and use Lemma 2.4. Since X/P X/h, this shows that the quantity ⌊ ⌋ j ≫ above is 2 2 2 2 2 Pj h X h k X X X h (28) 2 2 2 + 2k 2 2 + 2 . ≪ (log Pj) X P 2 X h Pj Pj ≪ (log Pj) j Xk 1119–13 Finally, if y log P , then Lemma 2.3 gives | | ≤ j iy X 10 (29) λ(m)m (log X)− , ≪ Pj X/P Xm 2X/P j+1≤ ≤ j iy say, so that bounding p p trivially by Pj/(log Pj) we see that this portion of ∈Pj ≪ the integral contributesP to Ij an amount 2 2 2 2 Pj X 20 h 2 19 − − (30) (log Pj) 2 2 (log X) 2 (log Pj) h (log X) . ≪ (log Pj) Pj X ≪ Combining this with (28), we obtain that I h2/(log P )2, and so from (27) it follows j ≪ j that X 2 2 2 2 h 1 2 h 2 h A(y) min 2 , 2 dy W ( ) max 2 W ( ) 9/5 . Z X | | X y ≪ P j (log Pj) ≪ P (log h) − Using this and (25) in (24), we conclude that 2X 2 1 1 1 Xh2 λ(n) dx Xh2 + + . Z ≪ (log h)9/5 W ( ) h2 ≪ log log h X x 4. THEOREM 1.1 – ROUND TWO We now refine the argument of the previous section, adding another ingredient which will permit us to obtain Theorem 1.1 in the substantially wider region h ≥ exp(10(log log X)10/9). Now we shall also need Lemmas 2.5 and 2.6. Let us suppose that h exp((log X)3/4), and let and be as in the previous ≤ P Pj section. Now we introduce a set of large primes consisting of the primes in the interval Q from exp((log X)4/5) to exp((log X)9/10). As with , let us also decompose into dyadic P Q intervals with denoting the primes in lying between Q =2k exp((log X)4/5) and Qk Q k Q =2k+1 exp((log X)4/5), where k runs from 0 to K (log X)9/10/ log 2. k+1 ∼ In place of (20) we now define the sequence a(n) by setting iy iy (31) A(y)= a(n)n = λ(p)p Aj(y), Xn Xj pX ∈Pj 1119–14 where iy iy (32) Aj(y)= λ(q)q λ(m)m . Xk qXk X/(P Q X) m 2X/(P Q ) ∈Q j+1 k+1 ≤ ≤ j k Now a(n) = 0 unless X/4 n 8X, and in the range X n 2X we have ≤ ≤ ≤ ≤ a(n) = λ(n)ω (n)ω (n), where ω (n) is as before, and ω analogously counts the P Q P Q number of prime factors of n in . Q As noted already in (22), a typical number in X to 2X will have ω (n) W ( ), and P ∼ P similarly will have ω (n) W ( )= q 1/q (1/10) log log X. Precisely, we have Q ∼ Q ∈Q ∼ P 1 1 ω (n)ω (n) W ( )W ( ) 2 XW ( )2W ( )2 + . P Q − P Q ≪ P Q W ( ) W ( ) X Xn 2X ≤ ≤ P Q Now set (analogously to (26)) X 2 2 2 iy 2 h 1 (33) Ij = (log Pj) p Aj(y) min 2 , 2 dy. Z X X | | X y − p j ∈P Then arguing exactly as in (24), (25), and (27), we find that 2X 2 X Xh2 (34) λ(n) dx max Ij + , Z ≪ W ( )2 j log log h X x iy Pj (35) j = y : log Pj y X, p 2 , E n ≤| | ≤ X ≥ (log Pj) o p j ∈P which denotes the exceptional set on which the sum over p does not exhibit ∈ Pj much cancelation. To bound the integral in (33) in the range log P y X, let us j ≤| | ≤ distinguish the cases when y belongs to the exceptional set , and when it does not. Ej Consider the latter case first, where by the definition of the sum over p does Ej ∈ Pj have some cancelation. So this case contributes to (33) 2 X 2 Pj 2 h 1 2 Aj(y) min 2 , 2 dy. ≪ (log Pj) Z X | | X y − The integral above is the mean value of a Dirichlet polynomial of size about X/Pj, which is larger than X/h. Therefore applying Lemma 2.4 (as in our estimate (28)) we obtain that the above is 2 2 2 2 Pj h X h 2 2 2 1 2 W ( ) . ≪ (log Pj) X Pj ≪ (log Pj) Q X/(2Pj )Xn 4X/Pj Xq n ≤ ≤ q | ∈Q 1119–15 Thus the contribution of this case to (34) is small as desired. Finally we need to bound the contribution of the exceptional values y : upon ∈ Ej bounding the sum over p trivially, this contribution to I is ∈ Pj j 2 2 2 2 h 1 2 h 2 (36) Pj Aj(y) min , dy Pj Aj(y) dy. ≪ Z | | X2 y2 ≪ X2 Z | | Ej Ej Now recall the definition of Aj(y) in (32), and use Cauchy-Schwarz on the sum over k (as in (26) or (33)) to obtain that the quantity in (36) above is (37) 2 2 2 2 2 h 2 iy iy Pj W ( ) max (log Qk) q λ(m)m dy. X2 k Z ≪ Q j X X E q k X/(Pj+1Qk+1) m 2X/(Pj Qk) ∈Q ≤ ≤ Since log Q (log X)4/5 (note 4/5 is bigger than 2/3+δ), in the range X y log P k ≥ ≥| | ≥ j we can use Lemma 2.3 to obtain π(Q ) 1 Q (38) qiy k+1 k , ≪ log Pj ≪ log Pj log Qk qX ∈Qk which represents a saving of 1/ log Pj over the trivial bound Qk/ log Qk. Using this in (37) and substituting that back in (36), we see that the contribution of the exceptional y to I is ∈Ej j 2 2 2 2 Pj W ( ) h 2 iy (39) Q max Qk λ(m)m dy. (log P )2 X2 k Z ≪ j j X E X/(Pj+1Qk+1) m 2X/(Pj Qk) ≤ ≤ It is at this stage that we invoke Lemmas 2.5 and 2.6. We are assuming that exp(10(log log X)10/9) h exp((log X)3/4), so that (log X)7 P Xǫ. Appeal- ≤ ≤ ≤ j ≤ ing to Lemma 2.5, it follows that X3/7+ǫ. Using now the bound of Lemma 2.6, |Ej|≪ we conclude that the quantity in (39) is P 2W ( )2 h2 X X h2W ( )2 j Q 2 3/7+ǫ 1/2+ǫ (40) 2 2 max Qk + X X Q 2 . ≪ (log Pj) X k PjQk PjQk ≪ (log Pj) Inserting these estimates back in (34), we obtain finally that 2X 2 Xh2 λ(n) dx , Z ≪ log log h X x Hal´asz-Montgomery Lemma 2.6 we need the measure of the exceptional set j to be a 1/2 E bit smaller than X , and to achieve this we needed Pj to be larger than a suitable power of log X. 1119–16 5. ONCE MORE UNTO THE BREACH Now we add one more ingredient to the argument developed in the preced- ing two sections, and this will permit us to obtain Theorem 1.1 in the range h > exp((log log log X)2). Moreover once this argument is in place, we hope it will be clear that a more elaborate iterative argument should lead to Matom¨aki and Radziwil l’sresult; we shall briefly sketch their argument, where the details are arranged differently, in the next section. Assume below that h exp(10(log log X)10/9). ≤ Let and be as in the previous section. Let (1) denote the set of primes lying P Q 1 9/10 1 P 9/10 between exp(exp( 100 (log h) )) and exp(exp( 30 (log h) )), so that this set is inter- mediate between and . Again split up (1) into dyadic blocks, which we shall index (1) P Q P as . In place of (31) we now define the sequence a(n) by setting Pj1 iy iy (41) A(y)= a(n)n = λ(p)p Aj(y), Xn Xj pX ∈Pj with iy (42) Aj(y)= λ(p1)p1 Aj,j1 (y), Xj1 X(1) p1 ∈Pj1 (1) where, with Mj,j1,k = X/(PjPj1 Qk), iy iy (43) Aj,j1 (y)= λ(q)q λ(m)m . Xk qXk M /8Xm 2M ∈Q j,j1,k ≤ ≤ j,j1,k Thus a(n) is zero unless n lies in [X/8, 16X] and on [X, 2X] we have a(n) = λ(n)ω (n)ω (1) (n)ω (n). P P Q Now arguing as in (33) and (34) we obtain 2X 2 X Xh2 (44) λ(n) dx max Ij + , Z ≪ W ( )2W ( (1))2 j log log h X x X 2+2ℓ P 2ℓ h2 1 (49) (P (1))9/5 piy j − A (y) 2 min , dy; j1 2 j,j1 2 2 Z X X (log Pj) | | X y − p j ∈P here ℓ is any natural number, and the inequality holds because on the sum over p Ej ∈ Pj is P /(log P )2 by assumption. We choose ℓ = (log P (1))/ log P . Now in (49), we ≥ j j ⌈ j1 j⌉ must estimate the mean square of the Dirichlet polynomial ( piy)1+ℓA (y), and p j j,j1 by our choice for ℓ this Dirichlet polynomial has length atP least∈P X, permitting an efficient use of Lemma 2.4. With a little effort, Lemma 2.4 can be used to bound (49) by (we have been a little wasteful in some estimates below) P 2ℓ h2 (2P )ℓX 2 (P (1))9/5 j − W ( )2(ℓ + 1)! j j1 2 2 (1) ≪ (log Pj) X Q Pj1 2 2 (1) 1/5 4ℓ 2 2 (1) 1/15 (50) W ( ) h (P )− (ℓ log P ) W ( ) h (P )− , ≪ Q j1 j ≪ Q j1 1119–18 (1) where at the last step we used log log Pj1 (1/30) log Pj. This contribution to (46) is ≤ (1) once again acceptably small (having saved a small power of Pj1 ), and completes the proof of Theorem 1.1 in this range of h. At this stage, all the ingredients in the proof of Theorem 1.1 are at hand, and one can begin to see an iterative argument that would remove even the very weak hypothesis on h made in this section! 6. SKETCH OF MATOMAKI¨ AND RADZIWIL L’S ARGUMENT FOR THEOREM 1.1 In the previous three sections, we have described some of the key ideas developed in [19]. The argument given in [19] arranges the details differently, in order to achieve quantitatively better results: our version saved a modest log log h over the trivial bound, and [19] saves a small power of log h. Instead of considering a(n) being λ(n) weighted by the number of primes in various intervals (as in Sections 3, 4, 5), Matom¨aki and Radziwil l deal with a(n) being λ(n) when n is restricted to integers with at least one prime factor in carefully chosen intervals (and a(n) = 0 otherwise). To illustrate, we revisit the argument in Section 3, and let be the interval defined there. Let denote the set of integers n [1, 2X] with n P S ∈ having at least one prime factor in . A simple sieve argument shows that there are P X/(log h)1/10 numbers n [X, 2X] that are not in . Therefore ≪ ∈ S 2X 2 2X 2 (51) λ(n) dx λ(n) + h 1 dx, Z ≪ Z X x 7. GENERALIZATIONS FOR MULTIPLICATIVE FUNCTIONS As mentioned in (14), the work of Matom¨aki and Radziwil lestablis hes short interval results for general multiplicative functions f with 1 f(n) 1 for all n. Our − ≤ ≤ treatment so far has been specific to the Liouville function; for example we have freely used the bounds of Lemma 2.3 which do not apply in the general situation. In this section we discuss an important special class of multiplicative functions (those that are “unpretentious”), and give a brief indication of the changes to the arguments that are needed. There is one notable extra ingredient that we need – an analogue of the Hal´asz-Montgomery Lemma for primes (see Lemma 7.1 below). A beautiful theorem of Hal´asz [8] (extending earlier work of Wirsing) shows that mean values of bounded complex valued multiplicative functions f are small unless f pretends to be the function nit for a suitably small value of t. When the multiplicative function is real valued, one can show that the mean value is small unless f pretends to be the function 1: this means that p x(1 f(p))/p is small. There is an extensive ≤ − literature around Hal´asz’s theorem andP its consequences; see for example [7, 9, 24, 37]. Let us state one such result precisely: suppose f is a completely multiplicative function taking values in the interval [ 1, 1], and suppose that − 1 f(p) (55) − δ log log X p ≥ pXX ≤ for some positive constant δ. Then uniformly for all t X and all √X x X2 we | | ≤ ≤ ≤ have x (56) f(n)nit , ≪ (log x)δ1 Xn x ≤ for a suitable constant δ1 depending only on δ. Now let us consider the analogue of Theorem 1.1 for such a completely multi- plicative function f, in the simplest setting of short intervals of length √X h ≥ ≥ 1119–21 exp((log X)17/18) (a range similar to that considered in Section 3). In this range we wish to show that 2X 2 (57) f(n) dx = o(Xh2), Z X x Xn Xj pXj X/P Xm 2X/P ∈P j+1≤ ≤ j so that a(n) is zero unless X/2 n 4X and in the range X n 2X we have ≤ ≤ ≤ ≤ a(n)= f(n)ω (n); all exactly as in (21). Now arguing as in (22)–(27) we obtain that P 2X 2 Xh2 (59) f(n) dx X max Ij + , Z ≪ j log log h X x 2 X 2 2 2 Pj iy h 1 h 2 f(m)m min 2 , 2 dy 2 , ≪ (log Pj) Z X X X y ≪ (log Pj) − X/Pj+1 m 2X/Pj ≤ ≤ which is acceptably small in (59). It remains to estimate the contribution to I from the exceptional set . Here we j Ej invoke the bound (56), so that the desired contribution is X2(log P )2 2 h2 1 (62) j f(p)piy min , dy. (log X)2δ1 P 2 Z X2 y2 ≪ j j X E p j ∈P Since h exp((log X)17/18) we have P exp((log h)9/10) exp((log X)17/20), and an ≥ j ≥ ≥ application of Lemma 2.5 shows that the measure of is exp((log X)1/6). This Ej ≪ is extremely small, and it is tempting to use the Hal´asz-Montgomery Lemma 2.6 to estimate (62). However this gives an estimate too large by a factor of log Pj, since Lemma 2.6 does not take into account that the Dirichlet polynomial in (62) is supported 1119–22 only on the primes. This brings us to the final key ingredient in [19] – a version of the Hal´asz-Montgomery Lemma for prime Dirichlet polynomials. Lemma 7.1.— Let T be large, and be a measurable subset of [ T, T ]. Then for any E − complex numbers x(p) and any ǫ> 0, 2 P log P x(p)pit dt + P exp x(p) 2. Z ≪ log P |E| − (log(T + P ))2/3+ǫ | | E pXP pXP ≤ ≤ it Proof. — We follow the strategy of Lemma 2.6. Put P (t) = p P x(p)p , and let I denote the integral to be estimated. Then using Cauchy-SchwarzP as≤ in (17), we obtain 2 2 2 p it (63) I x(p) 2 P (t)p− dt . ≤ X | | X − P Z p P p 2P E ≤ ≤ Now expanding out the integral above, as in (18), the second term of (63) is bounded by p i(t2 t1) P (t )P (t ) 2 p − dt dt . Z 1 2 − P 1 2 t1,t2 pX2P ∈E ≤ Now in place of (19), we can argue as in Lemma 2.3 to obtain p π(P ) (log P ) 2 pit + P exp , − P ≪ 1+ t 2 − (log(T + P ))2/3+ǫ pX2P ≤ | | where once again the small smoothing in the sum over p produces the saving of 1 + t 2 | | in the first term. Inserting this bound in (63), and proceeding as in the proof of Lemma 2.6, we readily obtain our lemma. Returning to our proof, applying Lemma 7.1 we see that the quantity in (62) may be bounded by 2 2 17/20 2 X (log Pj) Pj 1/6 (log X) Pj h + Pj exp (log X) . 2δ1 2 2/3+ǫ 2δ1 ≪ (log X) Pj log Pj − (log X) log Pj ≪ (log X) Thus the contribution of y to I is also acceptably small, and therefore (57) follows. ∈Ej j 8. SKETCH OF THE COROLLARIES We discuss briefly the proofs of Corollaries 1.5 and 1.6, starting with Corollary 1.5. The indicator function of smooth numbers is multiplicative, and so Matom¨aki and Radziwil l’s general result for multiplicative functions (see the discussion around (14)) shows the following: For any ǫ> 0 there exists H(ǫ) such that for large enough N the set = x [√N/2, 2√N] : the interval [x, x + H(ǫ)] contains no N ǫ-smooth number , E { ∈ } has measure ǫ√N. Now if for some x [√N, 2√N] we have x / and also |E| ≤ ∈ ∈ E N/x / , then we would be able to find N ǫ-smooth numbers in [x, x + H(ǫ)] and also in ∈E 1119–23 [N/x, N/x + H(ǫ)] and their product would be in [N, N +4H(ǫ)√N]. Thus if Corollary 1.5 fails, we must have (with χ denoting the indicator function of ) E E 2√N √N (χ (x)+ χ (N/x))dx 4 4ǫ√N, ≤ Z√N E E ≤ |E| ≤ which is a contradiction. Now let us turn to Corollary 1.6. First we recall a beautiful result of Wirsing (see [7], or [37]), establishing a conjecture of Erd˝os, which shows that if f is any real valued multiplicative function with 1 f(n) 1 then − ≤ ≤ 1 1 f(p) f(p2) lim f(n)= 1 1+ + + ... . N N − p p p2 →∞ nXN Yp ≤ The product above is zero if (1 f(p))/p diverges (this is the difficult part of p − Wirsing’s theorem), and is strictlyP positive otherwise. In Corollary 1.6, we are only interested in the sign of f and so we may assume that f only takes the values 0, 1. 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Cam- bridge Studies in Advanced Mathematics, 46, CUP (1995). Kannan SOUNDARARAJAN Stanford University Department of Mathematics 450 Serra Mall, Building 380 Stanford, CA 94305-2125, U.S.A. E-mail : [email protected]