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7 1 0 2 n a J 8 Note on the resonance method 1 for the Riemann zeta function ] T N Andriy Bondarenko and Kristian Seip . h t a To the memory of VictorHavin m [ Abstract. WeimproveMontgomery’sΩ-resultsfor ζ(σ+it) inthestrip 1/2 < σ < 1 and give in particular lower bounds|for the m|aximum of 1 ζ(σ+it) on √T t T that are uniform in σ. Wegive similar lower v b|oundsfo|rthemax≤imu≤mof P n−1/2−it onintervalsoflengthmuch 8 | n≤x | larger than x.Werely on our recent work on lower boundsfor maxima 7 9 of ζ(1/2+it) onlongintervals,aswellasworkofSoundararajan,Gál, | | 4 andothers.Thepaperaimsatdisplayingandclarifyingtheconceptually 0 different combinatorial arguments that show up in various parts of the . proofs. 1 0 Mathematics SubjectClassification (2010). 11M06, 11C20. 7 1 : v i X 1. Introduction r a Soundararajan[22] and Hilberdink [13] presented independently slightly different versionsof a technique, known as the resonance method, for detecting large values of the Riemann zeta function ζ(s). In our recent paper [6], we usedSoundararajan’sversionofthismethodandtheconstructionofaspecial multiplicative function to show that 1 1 logT logloglogT max ζ +it exp +o(1) (1.1) √T≤t≤T(cid:12)(cid:12) (cid:16)2 (cid:17)(cid:12)(cid:12)≥ (cid:16)√2 (cid:17)s loglogT ! when T (cid:12). This gav(cid:12)e an improvement by a power of √logloglogT com- →∞(cid:12) (cid:12) pared with previously known estimates [3, 22]. In this note, we will apply the resonance method to two closely related problems,namelytofindlargevaluesofrespectivelyζ(σ+it)for1/2<σ <1 and the partial sum n 1/2 it on certain long intervals (depending n M − − on M). We will find unif≤orm lower bounds on the maximum in the strip P ResearchsupportedinpartbyGrant227768oftheResearchCouncilofNorway. 2 Andriy Bondarenko and Kristian Seip 1/2<σ <1 andshowin particularthat the bound onthe right-handside of (1.1) (with1/√2 replacedby adifferentconstant)holds asfaras 1/loglogT to the right of the critical line. Before proceeding to the details of these new results, we would like to commentontherelationbetweenoursubjectandHardyspaces,thepresumed topicofthepresentvolume.Asoutlinedin[21],ourconstructionofresonators originates in Bohr’s several complex variables perspective of Dirichlet series and our recent work on Hardy spaces of Dirichlet series. The present paper can thus be viewedas an outgrowthof the remarkablyrich subject of Hardy spaces and more specifically of a branch of it that interacts with number theory.Moreover,onemayinterpretourtheoremonpartialsums(Theorem2 below) as dealing with a well known type of problem in complex analysis, namely how small the maximal size of an analytic function can be on a set of uniqueness that in some sense is “small”. For further information about HardyspacesofDirichletseriesandconnectionswithnumbertheory,werefer to the survey paper [21] and the monograph [18]. 2. Statement of main results A less precise version of the following result was stated without proof in [6]. Theorem1. There exists a positive and continuous function ν(σ) on (1/2,1), bounded below by 1/(2 2σ), with the asymptotic behavior − (1 σ) 1+O( log(1 σ)), σ 1 − ν(σ)= − | − | ր ((1/√2+o(1)) log(2σ 1), σ 1/2, | − | ց and such that the following holds. Ipf T is sufficiently large, then for 1/2+ 1/loglogT σ 3/4, ≤ ≤ (logT)1 σ − max ζ σ+it exp ν(σ) (2.1) t∈[√T,T](cid:12) (cid:16) (cid:17)(cid:12)≥ (cid:18) (loglogT)σ(cid:19) and for 3/4 σ 1 1(cid:12)/loglogT(cid:12), (cid:12) (cid:12) ≤ ≤ − (logT)1 σ − max ζ σ+it loglogT exp c+ν(σ) , (2.2) t [T/2,T] ≥ (loglogT)σ ∈ (cid:12) (cid:16) (cid:17)(cid:12) (cid:18) (cid:19) with c an absol(cid:12)ute consta(cid:12)nt independent of T. (cid:12) (cid:12) To place this result in context, we recall Levinson’s classical estimate1 [15] max ζ(1+it) eγloglogT +O(1), (2.3) 1 t T| |≥ ≤ ≤ where γ is the Euler–Mascheroniconstant. We now observe that Theorem 1 gives a “smooth” transition between the two endpoint cases (2.3) and (1.1). The factor loglogT onthe right-handside of (2.2) is only needed for σ close to the right endpoint 1 1/loglogT, to get the transition to Levinson’s − 1ThisresultwaslaterimprovedbyGranvilleandSoundararajan[12]whomanagedtoadd apositivetermofsizelogloglogT ontheright-handsideof (2.3). Note on the resonance method for the Riemann zeta function 3 estimate. Theorem 1 gives a notable improvement of a classical estimate of Montgomery [17] for the range 1/2 < σ < 1. See [19] and the discussion in [6] for the best estimates known previously. Thechoiceofintermediateabscissaσ =3/4issomewhatarbitrary(any fixed σ , 1/2<σ <1 would do), and we could have shortened the interval 0 0 in (2.1) (depending on σ). Indeed, the precise statement of Theorem 1 is a tradeoffbetweenconveyingthe mainpoint ofthe transitionbetween the two endpoint cases and keeping the technicalities reasonably simple. We have refrained from making a precise statement about sharp estimates in the short intervals [1/2,1/2+1/loglogT] and [1 1/loglogT,1], − although our method would certainly allow us to do it. The main point is that the order of magnitude of the respective endpoint estimates persists in these intervals. It may seem surprising that these intervals are as long as 1/loglogT oneitherside.Wewillseeinthecourseofthe proofthatthiscan beattributedtothe resonancemethod’s selectionofsmoothnumbers2 inthe construction of resonating Dirichlet polynomials. In our proof of Theorem 1, we use the approximate formula x1 σ it ζ(σ+it)= n σ it − − +O(x σ), (2.4) − − − − 1 σ it n x − − X≤ which holds uniformly in the range σ σ > 0, t x (see [23, Theorem 0 ≥ | | ≤ 4.11]). This means that detecting large values of ζ(σ+it) for 1/2 σ 1 ≤ ≤ and t T is mainly a question about finding large values of the Dirichlet | | ≤ polynomial n σ it for t T. n T − − | |≤ We find it≤to be of interest to see what we get when we look for large P values of just the partial sum itself on longer intervals. Thus we remove the a priori restriction on the length of the interval forced upon us by the approximate formula (2.4). We will only consider σ =1/2 and introduce the notation DM(t)= n−1/2−it. n M X≤ Theorem 2. Suppose that c, 0 < c < 1/2, is given. If T is sufficiently large and M exp e logT loglogT logloglogT/2 , then ≥ (cid:0) p logT(cid:1)logloglogT max D (t) exp c . M t [√T,T]| |≥ s loglogT ! ∈ This theorem gives information about the precision of the resonance methodaswellasitslimitations.Wenoticethattheglobalmaximumsatisfies D :=max D (t) √M, M M k k∞ t | |∼ and hence we see that our method gives us that when M takes the minimal value exp e logT loglogT logloglogT/2 , the maximum on [√T,T] is at 2The smoot(cid:0)hnpess (sometimes called the friability)(cid:1)of a positive integer n is measured by thelargestprimepdividingn.Thesmallerthisprimeis,thesmootherthenumberis. 4 Andriy Bondarenko and Kristian Seip least D η/loglogM for somepositive number η. This means thatthe value M k k∞ of the maximum “predicted” by the resonance method is at most a power of order1/loglogT (orequivalentlyoforder1/loglogM)offthetruemaximum (whateveritis).Ontheotherhand,whileweknowthat D (t) “eventually” M | | will come arbitrarily close to the absolute maximum, the interval [√T,T] is of course far too short for us to guarantee, by general considerations, that wegetanywherenear D .Hence the resonancemethodcouldinfactbe M k k∞ considerablymore precise than what we cansafely conclude that it is in this case. A word on notation, before we turn to a general discussion of the resonance method and the proofs of our theorems: We will use the shorthand notation log x:=loglogx and log x:=logloglogx. 2 3 3. The resonance method—general considerations The basic idea of either versions of the resonance method is to identify a special Dirichlet polynomial R(t)= r(m)m it − m X∈N that“resonateswell”withtheobjectathand,whichinourcaseisthepartial sum n σ it onagiveninterval.Theprecisemeaningofthisisthatthe n x − − integral≤of R(t)2 (mollified by multiplication by a suitable smooth bump P | | function) times n σ it is as large as possible, given that the coeffi- n x − − cients r(m) have a fi≤xed square sum and also subject to whatever a priori P restrictions we need to put on the set of integers . The method will not N onlyproducelargevalues,butalsogiveinformationaboutwhichoftheterms in n σ it that contribute in an “essential” way. n x − − T≤he technicalities will differ considerably depending on σ, for reasons P thatwillbecomeclearbelow.Inourstudyofζ(σ+it)intherange3/4 σ ≤ ≤ 1, we will use Soundararajan’s original method. This means that we choose a smooth function Ψ compactly supported in [1/2,1], taking values in the interval [0,1] with Ψ(t)=1 for 5/8 t 7/8. We define ≤ ≤ M (R,T):= ∞ R(t)2Ψ t dt, (3.1) 1 | | T Z−∞ (cid:16) (cid:17) M (R,T):= ∞ ζ(σ+it)R(t)2Ψ t dt. (3.2) 2 | | T Z−∞ (cid:16) (cid:17) Then M (R,T) 2 max ζ(σ+it) | |, (3.3) T/2 t T ≥ M1(R,T) ≤ ≤ (cid:12) (cid:12) and the goal is therefore to m(cid:12) aximize(cid:12)the ratio on the right-hand side of (3.3). We require that max T1 ε for some fixed ε, 0 < ε < 1, and get − N ≤ Note on the resonance method for the Riemann zeta function 5 by straightforwardcomputations (see [22, pp. 471–472])that M (R,T)=TΨˆ(0) 1+O(T 1) r(n)2 (3.4) 1 − | | n (cid:0) (cid:1) X∈N and r(m)r(n) M (R,T)=TΨˆ(0) +O(T1 σlogT) r(n)2. (3.5) 2 kσ − | | n ,mk=n n ∈NX X∈N Hence the problem of estimating the right-hand side of (3.3) boils down to finding out how large the ratio r(m)r(n) r(n)2 (3.6) kσ | | n∈NX,mk=n .nX∈N can be under the a priori restriction that max T1 ε. This problem was − N ≤ solved in [22, Theorem 2.1] for σ =1/2. Asshownin[6],wecandobetterwhenσ =1/2byremovingtheapriori restriction that max T1 ε, and this is also true when σ is not too close − N ≤ to1.Ifweagainmanagetoreducethe problemtothatofmaximizingaratio like the one in (3.6),then we clearlyarein a better position. However,arriv- ing at such an optimization problem is less straightforward, mainly because more terms will contribute in either of the sums representing respectively M (R,T) and M (R,T). An additional problem is that sets of integers 1 2 N involved in making expressions like (3.6) large typically enjoys a multiplicative structure, while estimating sums like those representing M (R,T) and 1 M (R,T)requiressome“additivecontrol”.Wewillnowpresenttheremedies, 2 introduced in [6], for getting around these problems. We begin with the problem of “additive control”. We go “backwards” and start from the problem of maximizing f(m)f(n) f(n)2 (3.7) kσ | | n∈MX,mk=n .nX∈M for a suitable set and arithmetic function f(n) under the condition that M N. We then extract the resonating Dirichlet polynomial from a so- |M| ≤ lution (or approximate solution) to this problem as follows, assuming as we maythatf(n)isnonnegative.Followinganideafrom[1],welet bethe set J of integers j such that (1+T 1)j,(1+T 1)j+1 = , − − M6 ∅ and let m be the mh inimum of (1+T 1)j,((cid:17)1\+T 1)j+1 for j in . j − − M J Then set (cid:2) (cid:1)T := m : j j N ∈J and (cid:8) (cid:9) 1/2 r(m):= f(n)2 (3.8)   n∈M,1−T−1(logT)X2≤n/m≤1+T−1(logT)2   6 Andriy Bondarenko and Kristian Seip for every m in . Note that plainly N. By taking the local ℓ2 N |N|≤|M|≤ average as in (3.8), we get a precise relation between f(n) and r(n), while at the same time we get the desired “additive control” because each of the intervals (1+T 1)j,(1+T 1)j+1 contains at most one integer from . − − N We turn nextto the counterpartsto (3.1) and(3.2). We considernow a (cid:2) (cid:1) longerintervaloftheform[Tβ,T]with0<β <1;itwillbeconvenientforus tofixonceandforallβ =1/2.Moreover,weusetheGaussianΦ(t):=e t2/2 − as mollifier. Our replacements for (3.1) and (3.2) are then, respectively, logT M (R,T):= R(t)2Φ t dt, 1 | | T Z√T≤|t|≤T (cid:16) (cid:17) Mf(R,T):= ζ(σ+it)R(t)2Φ logTt dt, (3.9) 2 | | T Z√T≤|t|≤T (cid:16) (cid:17) and we get tfhat M (R,T) 2 max ζ(σ+it) | |. (3.10) √T≤t≤T(cid:12) (cid:12)≥ Mf1(R,T) We state first the estima(cid:12)te for M (cid:12)(R,T) obtained in [6, Formula (22)]. 1 f This is a matter of direct computation based on the definitions given above. f Lemma 3. There exists an absolute constant C such that M (R,T) CT(logT)3 f(n)2. (3.11) 1 ≤ n X∈M To estimate (3.9f), we extend the integral to the whole real line, so that wecantakeadvantageofthefactthattheFouriertransformΦofΦispositive. We chose the larger set √T t T and a different dilation factor of the ≤ | | ≤ mollifier ((logT)/T instead of 1/T), because these choicesballow us to get the control we need of the integral over the complementary set. Indeed, the estimation for t T is trivial because of the rapid decay of the Gaussian | | ≥ andourchoiceofdilationfactor,whilethefollowingestimatetakescareofthe interval t √T: For arbitrary numbers λ > 0, 0 < β < 1, and 0 < σ < 1, | | ≤ we have Tβ λ it logT n σ Φ t dt Cmax Tβ,M1 σlogM , − − n T ≤ (cid:12)(cid:12)1≤Xn≤M Z−Tβ(cid:18) (cid:19) (cid:16) (cid:17) (cid:12)(cid:12) (cid:0) (cid:1)(3.12) (cid:12) (cid:12) where the constant C is independent of λ, β, σ. This is proved by making a minor adjustment of the proof of [6, Lemma 4], which deals only with the caseσ =1/2.Doingthe samecomputationsasin[6],relyingcruciallyonthe positivity of φ, we arrive at the following lemma (see formula (14) in [6]): Lemma4. Suppose1/2 σ 1and √T.There exist absolutepositive b ≤ ≤ |M|≤ constants c,C such that T f(n)f(m) M (R,T) c CT(logT)4 f(n)2 . 2 ≥ logT kσ −  n ,mk=n,k T n ∈M X ≤ X∈M f   Note on the resonance method for the Riemann zeta function 7 The powers of logT are harmless if σ σ < 1 for some fixed σ , but 0 0 ≤ they make this lemma useless when σ is close to 1 since the lower bound in (2.3) is of order log T. This is why we need both versions of the resonance 2 methodwhenconsideringthewholerange1/2+1/log T σ 1 1/log T. 2 ≤ ≤ − 2 The resonance method for the partial sum problem yields the same bounds, up to an obvious modification of the indices in the summation in Lemma 4. Indeed, defining logT M (R,T):= D (t)R(t)2Φ t dt, 2 M | | T Z√T≤|t|≤T (cid:16) (cid:17) we get: ff Lemma 5. Suppose that √T. There exist absolute positive constants |M| ≤ c,C such that T f(n)f(m) M (R,T) c CT(logT)4 f(n)2 . 2 ≥ logT √k −  n ,mk=n,k M n f ∈M X ≤ X∈M f   We are now left with the problem of making the first sum on the right- hand side large; the problem of making the right-hand side of (3.7) large is just the special case when M =[T].In the next session,we will show how to deal with this problem for a wide range of values of M. 4. Gál-type extremal problems and proof of Theorem 1 4.1. Background on Gál-type extremal problems Before turning to the extremal problems arrived at in the previous section, wewouldliketoplacethemincontextbydescribingbrieflyaline ofresearch that has been instrumental for our approach. This is the study of greatest common divisor (GCD) sums of the form (m,n)2σ (4.1) (mn)σ m,n X∈M and the associated (normalized) quadratic forms (m,n)2σ f(m)f(n) f(n)2, (4.2) (mn)σ m,n n X∈M (cid:14) X∈M where isasaboveandweagainassumethatf(n)isnonnegativeanddoes M not vanish on . We observe that (3.7) is smaller than (4.2) because the M former is obtained from the latter by restricting the sum in the nominator to a subset of . In most cases of interest when 1/2 σ <1, we may M×M ≤ obtain a reverse inequality so that the two expressionsare of the same order of magnitude. In general, it is clear that if (4.2) is large, then also (3.7) will be large. Theproblemistodecidehowlargeeitherofthetwoexpressions(4.1)or (4.2) can be under the assumption that = N, and more specifically we |M| 8 Andriy Bondarenko and Kristian Seip areinterestedintheasymptoticswhenN andσisfixedwith0<σ 1. →∞ ≤ We refer to (4.1) and (4.2) as Gál-type sums because the topic begins with a sharp bound of Gál [11] (of order CN(loglogN)2) for the growth of (4.1) when σ =1. Dyer and Harman [9] obtained the first nontrivial estimates for the range 1/2 σ < 1, and during the past few years, we have reached an ≤ essentially complete understanding for the full range 0 < σ 1, thanks to ≤ the papers [2, 4, 5, 16]. The techniques used for different values of σ differ considerably,andtheproblemisparticularlydelicateforσ =1/2atwhichan interesting“phasetransition”occurs.Wereferto[21]foranoverviewofthese results and to [2, 16] for information about the many different applications of such asymptotic estimates. It is the insight accumulated in this research that has led to the con- structions given below. More specifically, we will follow Gál [11] when σ is close to 1 and [6] when σ is close to 1/2. The reader will notice that our set will contain very smooth numbers when σ is close to 1 in contrast M to what happens near σ = 1/2. Our treatment of the latter case shows that more and more primes are needed when σ decreases; the simplest possible choice (made by Aistleitner in [1]) of taking r to be of size logN/log2 and the n to be the divisors of the square-free number p p will be nearly j 1 r ··· optimal only when σ is “far” from the endpoints 1 and 1/2.Translating this philosophy to Soundararajan’s method, we find that the terms picked out in the approximating sum n σ it correspond to decreasingly smooth n T − − numbers when σ goes from 1 t≤o 1/2. P 4.2. Levinson’s case σ =1 revisited It is instructive to consider first the endpoint case σ =1. We will now show that max ζ(1+it) eγlog T +O(log T). (4.3) T/2 t T| |≥ 2 3 ≤ ≤ This estimate is slightly worse than (2.3) and the best known result of Granville and Soundararajan [12], but the benefit is the simplicity of the proof and also that the interval has been shortened. We notice at this point thatHilberdinkgottheestimate(2.3)byhisversionoftheresonancemethod. We fix a positive number x and an integer ℓ (to be determined later) and let be the set of divisors of the number M K =K(x,ℓ):= pℓ 1. − p x Y≤ We require that K [√T] and choose r(n) to be the characteristic function ≤ of . A computation shows that M ℓ 1 1 − ℓ ν = ℓ+ − . kσ pνσ n∈MX,mk=n pY≤x(cid:16) νX=1 (cid:17) Note on the resonance method for the Riemann zeta function 9 Hence ℓ 1 r(m)r(n) − ν r(n)2 = 1+ 1 p νσ . (4.4) − kσ − ℓ mXk=n .nX∈M pY≤x(cid:16) νX=1(cid:16) (cid:17) (cid:17) We now set σ =1 and find that ℓ 1 r(m)r(n) − ν r(n)2 = (1 p 1) 1+ p ν +O(p ℓ) − − − − k − ℓ mXk=n .nX∈M pY≤x(cid:16) νX=2 (cid:17) =(1+O(ℓ−1)) (1 p−1)−1 − p x Y≤ 1 = 1+O(ℓ 1)+O eγlogx, − √xlogx (cid:18) (cid:16) (cid:17)(cid:19) where we in the last step used Mertens’s third theorem (see [8] for a precise analysis of the error term). By the prime number theorem, we may choose x = (logT)/(2log T) and ℓ = [log T] if T is large enough. Taking into 2 2 account (3.4) and (3.5), we obtain the desired result (4.3). 4.3. The case 3/4 σ 1 1/log T ≤ ≤ − 2 We follow the argument of the preceding subsection up to (4.4), from which we deduce that r(m)r(n) 1 r(n)2 1+ 1 p σ − kσ | | ≥ − ℓ mXk=n .nX∈M pY≤x(cid:16) (cid:16) (cid:17) (cid:17) 1 1 1+p σ −ℓ; (4.5) − ≥ pY≤x(cid:16) (cid:17) herewe usedBernoulli’sinequality inthe laststep. We willuse the following lemma to estimate the latter expression. Lemma 6. There exists an absolute constant C such that x1 σ p−σ ≥σlog2x+C+ (1 σ−)logx p x − X≤ whenever (1 σ)logx 1/2. − ≥ Proof. By Abel summation and the inequality π(x)>x/logx which is valid for x 17 (see [20]), we find that ≥ x1 σ x dy p σ − +σ +C , (4.6) − ′ ≥ logx yσlogy p x Z2 X≤ 10 Andriy Bondarenko and Kristian Seip where C is an absolute constant. Making the changeof variablesu=log y, ′ 2 we see that x dy = log2xe(1 σ)eudu − yσlogy Z2 Zlog22 ∞ (1 σ)j(logjx logj2) =log x log 2+ − − 2 − 2 j j! j=1 · X (1−σ)logx ey 1 =log x log 2+ − dy. 2 − 2 y Z(1−σ)log2 Now using the trivial bound a ey 1 ea a+1 − dy y ≥ a − a Z0 and returning to (4.6), we obtain the desired estimate. (cid:3) We are now prepared to give the first part of the proof of Theorem 1. Proof of Theorem 1—part 1. Makingthesamechoicesx=(logT)/(2log T) 2 and ℓ = [log T] as in the case σ = 1 and returning to (4.5), we see that 2 Lemma 6 gives that r(m)r(n) r(n)2 kσ | | mXk=n .nX∈M 2σ 1(logT)1 σ − − exp σlog T + E(T,σ) , ≥ 3 (1 σ)(log T)σ − (cid:18) − 2 (cid:19) where (1+δ)log T (logT)1 σ E(T,σ) C+ 3 − ≤ (1 σ)(log T)σ+1 − 2 forarbitraryδ >0whenT issufficiently large.Returningto(3.3),(3.4),and (3.5),we nowobtain(2.2) andthe desiredasymptoticbehaviorofν(σ) when σ 1 because ր log T 3 (1 σ) log(1 σ) log T ≤ − | − | 2 when log T e, by our a priori assumption that σ 1 1/log T. We also 2 ≥ ≤ − 2 get the uniform lower bound ν(σ) 1/(2 2σ) because 2σ 1 > 1/2 when − σ 1/2 and log T/log T 0 whe≥n T − . (cid:3) ≥ 3 2 → →∞ 4.4. The case 1/2+1/log T σ 3/4 2 ≤ ≤ In view of the preceding section, we already know that (2.2) holds for large T when we choose ν(σ) to be an appropriate function bounded below by 1/(2 2σ). This is just because the interval [T/2,T] is shorter than [√T,T] − when T 4. What remains is therefore to prove that ν(σ) can chosen such ≥ that it also has the desired asymptotic behavior when σ 1/2, while (2.2) ց still holds for large T.