SOME OPENQUESTIONS IN ANALYSISFOR DIRICHLETSERIES EEROSAKSMANANDKRISTIANSEIP ABSTRACT. Wepresentsomeopenproblemsanddescribebrieflysomepossibleresearchdi- rectionsintheemergingtheoryofHardyspacesofDirichletseriesandtheirintimatecounter- parts,Hardyspacesontheinfinite-dimensionaltorus.Linkstonumbertheoryareemphasized 6 1 throughoutthepaper. 0 2 n a 1. INTRODUCTION J Wehaveinrecentyearsseenanotablegrowthofinterestincertainfunctionalanalyticas- 7 pectsofthetheoryofordinaryDirichletseries ] A ∞ a n s. F n − . n 1 h X= t Contemporaryresearchinthisfieldowes muchtothefollowingfundamentalobservationof a H.Bohr[20]: Bythetransformationz p s (herep isthe jthprimenumber)andthefun- m j = −j j damentaltheoremofarithmetic,anordinaryDirichletseriesmaybethoughtofasafunction [ 1 ofinfinitelymanycomplexvariablesz1,z2,.... Moreprecisely,intheBohrcorrespondence, v 6 (1) F(s): ∞ a n s f(s): a zν, n − ν 61 =nX=1 ∼ =ν∈XN∞fin 1 wheren pν1 pνk andweidentifya withthecorrespondinegcoefficienta ,andN stands 0 = 1 ··· k ν n ∞fin . for the finite sequences of positive indices. By a classical approximation theorem of Kro- 1 0 necker, this is much more than justea formal transformation: If, say, only a finite number 6 of the coefficients a are nonzero (so that questions about convergence of the series are n 1 avoided),thesupremumoftheDirichletpolynomial a n s inthehalf-planeRes 0equals v: the supremum of the corresponding polynomial on thne in−finite-dimensional pol>ydisc D . ∞ i P X In a groundbreaking work of Bohnenblust and Hille [19], it was later shown that homoge- r neous polynomials—the basic building blocks of functions analytic on polydiscs—may, via a themethodofpolarization,betransformedintosymmetricmultilinearforms. Bohnenblust andHilleusedthisinsighttosolvealong-standingprobleminthefield:Bohrhadshownthat the width of the strip in which a Dirichlet series converges uniformly but not absolutely is 2000MathematicsSubjectClassification. 11C08,11C20, 11M06,11N60, 32A05,30B50, 42B30,46B09, 46B30, 46G25,47B35,60G15,60G70. SaksmanissupportedbytheFinnishCoEinAnalysisandDynamicsResearchandbyaKnutandAliceWal- lenbergGrant.SeipissupportedbyGrant227768oftheResearchCouncilofNorway. 1 2 EEROSAKSMANANDKRISTIANSEIP 1/2, but Bohnenblustand Hillewere able to prove thatthis upperestimateis in fact opti- ≤ mal. Inretrospect,onemayintheworkofBohrandBohnenblust–Hilleseetheseedsofatheory ofHardy Hp spacesofDirichletseries. However,thisresearchtookplacebeforethemodern interplaybetween function theoryand functional analysis, as well as theadvent of the field of several complex variables, and the area was in many ways dormant until the late 1990s. Oneof themaingoalsof the1997paperofHedenmalm, Lindqvist,and Seip[48] was to ini- tiateasystematicstudyofDirichletseriesfromthepointofviewofmodernoperator-related functiontheoryandharmonicanalysis.Independently,atthesametime,apaperofBoasand Khavinson [18] attracted renewed attention, in the context of several complex variables, to theoriginalworkofBohr. Themainobjectofstudyin[48]istheHilbertspaceofDirichletseries a n s withsquare n n − summablecoefficients a . This Hilbert space H2 consists of functions analytic in the half- n P planeRes 1/2. Itsreproducingkernelat s isk (w) ζ(s w),whereζistheRiemannzeta s > = + function. BytheBohrcorrespondence,H2 maybethoughtofastheHardyspace H2 onthe infinite-dimensionaltorus T . Bayart [10] extended the definitionto any p 0 by defining ∞ > Hp astheclosureofDirichletpolynomialsF(s) N a n s underthenorm = n 1 n − = 1 TP 1/p F Hp : lim F(it) pdt . k k = ³T→∞2T Z−T| | ´ By ergodicity (or see [79] for an elementary argument), the Bohr correspondence yields the identity 1/p (2) F Hp f Hp(T ): f(z) pdm (z) , k k =k k ∞ = T | | ∞ ³Z∞ ´ wherem standsfortheHaarmeasureonthedistinguishedboundaryT ,i.e.,fortheprod- ∞ uct of co∞untably many copies of normalized Lebesgue measure on the circle T. Since the HardyspacesontheinfinitedimensionaltorusHp(T )maybedefinedastheclosureofana- ∞ lyticpolynomialsintheLp-normonT ,itfollowsthattheBohrcorrespondenceprovidesan ∞ isomorphismbetweenthespacesHp(T )andHp. Thislinearisomorphismisbothisomet- ∞ ricandmultiplicative. TheclassicaltheoryofHardyspacesandtheoperatorsthatactonthemservesasanimpor- tantsourceofincitementforthefieldofDirichletseriesthathasevolvedafter1997. Twodis- tinctfeaturesshouldhoweverbenoted. First,anumberofnewphenomena,typicallycross- ingexistingdisciplines,appear thatare notpresentin theclassical situation. Second, many oftheclassicalobjectschangeradicallyandrequirenewviewpointsandmethodsinorderto beproperlyunderstoodandanalyzed. In thefollowingsections, we sketchbrieflysomeresearch directionsandlistseveral open problems (thus updating [47]). In our selection of problems, we have followed our own in- terests and made no effort to compile a comprehensive list. As a consequence, several in- terestingrecentdevelopmentssuchasforinstance[13]or[67]willnotbeaccountedforand discussed. The reader should also notice that the difficulty of the problems may vary con- siderably. It seems likely that for some of the problems mentioned below, further progress SOMEOPENQUESTIONSINANALYSISFORDIRICHLETSERIES 3 willrequirenovelandunconventionalcombinationsoftoolsfromharmonic,functional,and complexanalysis,aswellasfromanalyticnumbertheory. 2. BASIC PROPERTIES OF THE SPACES Hp AND Hp(T∞) ThestudyoftheboundarylimitfunctionsinthespacesHphasanumberofinterestingfea- tures. Severalcentralpointshavebeen clarified,such as questionsconcerningconvergence oftheDirichletseries[49],towhatextentergodicityextendstotheboundary[79],properties oftheboundarylimitfunctionsforDirichletseriesinH2 [71],andzerosoffunctionsinH2 and, at least partially, in Hp for p 2 [82]. The diversity of techniques involved is consid- > erable, ranging from function theory in polydiscs and ergodic theory to classical harmonic analysis, Hardy space techniques, Fourier frames, estimates for solutions of the ∂ equation, andRamanujan’sestimatesforthedivisorfunction. Still,averynaturalproblemfirstconsid- eredin[10](see[79]forfurtherdiscussiononit)remainsunsolvedandrepresentsoneofthe mainobstaclestofurtherprogress: Problem2.1(Theembeddingproblem). IstheLpintegralofaDirichletpolynomial N a n s n 1 n − over any segmentof fixed lengthon theverticallineRes 1/2 boundedby a univers=alcon- stanttimes N a n s p ? = P k n 1 n − kHp = ThisisknownPtoholdforp 2andthustriviallyforp aneveninteger.Onemaynoticeacuri- = ousresemblancewithMontgomery’sconjecturesconcerningnorminequalitiesforDirichlet polynomials(see[65,pp. 129,146]or[56,p. 232–235]). Itremainstobeclarifiedifthereisa linkbetweenthisquestionandMontgomery’sconjectures. An affirmativeanswer to Problem 2.1 for p 2 would have immediate function theoretic < consequences regarding for instancezero sets and boundarylimits. Namely, following [71], wewouldbeabletoanswer Problem2.2. CharacterizeCarlesonmeasuresforHp on{Res 1/2}forp 2. > < Moremodestbutnontrivialopenquestionsare: Problem2.3. DothezerosetsoffunctionsinHp for p 2satisfytheBlaschkeconditionin < thehalf-planeRes 1/2? > Problem2.4. AreelementsofHp forp 2locallyintheNevanlinnaclass? < TherearesimilarproblemsofadualflavorregardinginterpolatingsequencesforHp. In- deed, it follows from [72] that theShapiro–Shieldsversionof Carleson’sclassical theorem in thehalf-planeRes 1/2remainsvalidwhen1/p isaneveninteger. Wewouldliketoknowif > thisresultextendstoothervaluesofp. ByatheoremofHelson[50],thepartialsumoperator[50]isuniformlyboundedonHp for 1 p (see[4]foranalternativetreatment),andhencethefunctionsn s forn 1forma − < <∞ ≥ basisforHp fortheseexponentsp. Thefollowingquestionsstatedin[4]seemtobeopen: Problem2.5. DoesHp haveanunconditionalbasisifp (1, )andp 2? ∈ ∞ 6= Problem2.6. DoesH1haveabasis? Doesithaveanunconditionalbasis? 4 EEROSAKSMANANDKRISTIANSEIP Thelasttwo problemsare equivalentto correspondingstatementsfor Hp(T ). Thereare ∞ alsonaturalandinterestingquestionsthatarespecificforfunctiontheoryininfinitedimen- sions. In[5](see[79]forthefirststepsinthisdirection),itwas shownthatFatouorMarcin- kiewicz–Zygmund-type theorems on boundary limits remain true for all classes Hp(T ) or ∞ for theirharmoniccounterpartshp(T ), assumingfairlyregular radial approach to the dis- ∞ tinguishedboundaryT ;thesimplestexampleofsuchapproachisoftheform(reiθ1,r2eiθ2, ∞ r3eiθ3,...) with r 1 . However, [5] also constructs an example of an element f in H (T ) − ∞ ∞ ↑ suchthatatalmosteveryboundarypoint, f failstohavearadiallimitunderacertainradial approachthatisindependentoftheboundarypoint. Problem2.7. Givegeneralconditionsforaradial(ornon-tangential)approachinD toT ∞ ∞ suchthatFatou’stheoremholdsforelementsinHp(T ). ∞ The Hp spaces are well defined (via density of polynomials) also in the range 0 p 1. < < Again,onemayinquiretheanalogueoftheembeddingproblem(nowstatedintermoflocal Hardy spaces on Res 1/2). For all values other than p 2, even partial non-trivialresults = = pertainingtothefollowingwidelyopenquestion(see[79])wouldbeinteresting. Problem2.8. DescribethedualspacesofHp. 3. OPERATOR THEORY AND HARMONIC ANALYSIS Viewing our Hardy spaces as closed subspaces of the ambient Lp spaces on the infinite- dimensionaltorusT ,weareledtoconsiderclassicaloperatorsliketheRieszprojection(or- ∞ thogonalprojection from L2 to H2), Hankel operators, and Fouriermultiplieroperators. [4] contains some results on multipliers and Littlewood–Paley decompositions. It has become clear, however, that most of the classical methods are either not relevant or at least insuffi- cient for the infinite-dimensional situation. For example, the classical Nehari theorem for Hankel forms (or small Hankel operators)does not carry over to T , see [73]. This leads us ∞ toaskifareasonablereplacementcanbefoundand,moregenerally,howthedifferentroles and interpretations of BMO (the space of functions of bounded mean oscillation) manifest themselvesinourinfinite-dimensionalsetting. Problem3.1. WhatisthecounterparttoNehari’stheoremonT ? Inparticular,whatcanbe ∞ saidabouttheRieszprojectionofL (T )andotherBMO-typespacesonT ? ∞ ∞ ∞ This and similar operator theoretic problems may be approached along several different paths. In [29], a natural analogueof the classical Hilbert matrixwas identified and studied. This matrix was referred to as the multiplicative Hilbert matrix because its entries a : m,n = (pmnlog(mn)) 1 depend on the product m n. This matrix represents a bounded Hankel − · formonH2 H2withspectralproblemssimilartothoseoftheclassicalHilbertmatrix.(Here 0× 0 H2denotesthesubspaceofH2consistingoffunctionsthatvanishat .)Itsanalyticsym- 0 +∞ bolϕ isaprimitiveof ζ(s 1/2) 1,andbyanalogywiththeclassicalsituation,weareled 0 − + + tothefollowingproblem. Problem3.2. Isthesymbolϕ (s) 1 (logn) 1n 1/2 s theRieszprojectionofafunction 0 = + ∞n 2 − − − inL (T )? = ∞ ∞ P SOMEOPENQUESTIONSINANALYSISFORDIRICHLETSERIES 5 ItisinterestingtonoticethatapositiveanswertoProblem2.1for p 1wouldyieldapos- = itiveanswer tothisquestion,viaan argumentinvolvingCarlesonmeasures. Wereferto[29] fordetails. ThebeautifulpioneeringcontributionofGordonandHedenmalm[42]andagrowingnum- berofotherpapershaveestablishedthestudyofcompositionoperatorsonHardyspaces of Dirichlet series as an active research area in the interface of one and several complex variables. Intheseriesofpapers[77,16,12],quantitativeandfunctionalanalytictoolshavebeen developedinthiscontext,forexamplenormestimatesforlinearcombinationsofreproduc- ing kernels, Littlewood–Paley formulas, and (soft) functional analytic remedies for the fact thatHp failstobecomplementedwhen1 p andp 2. ≤ <∞ 6= Problem3.3. CharacterizethecompactcompositionoperatorsonH2. 4. MOMENTS OF SUMS OF RANDOM MULTIPLICATIVE FUNCTIONS Therehasduringthelastfewyearsbeenaninterestinginterplaybetweenthestudyofsums of random multiplicativefunctions and problems and methodscoming from Hardy spaces. Thistopichasalonghistory,beginningwithanimportpaperofWintner[84]. Oneofthelinks toHardyspacescomesfrom Problem4.1(Helson’sproblem[52]). Isittruethatk nN 1n−skH1 =o(pN)whenN →∞. = This intriguing open problem arose from Helson’sPstudy of Hankel forms and a compar- ison with the one-dimensional Dirichlet kernel. However, it seems to be more fruitful to thinkoftheprobleminprobabilisticterms,viewingthefunctionsp s asindependentStein- −j hausvariables. Resortingtoadecompositionintohomogeneouspolynomialsandusingwell knownestimatesforthearithmeticfunctionΩ(n),itwasshownin[26]thatk nN 1n−skH1 ≫ pN(logN) 0.05616.ThiswaslaterimprovedbyHarper,Nikeghbali,andRadziwiłł=[45]who,us- − P ingmethodsfrom[44],foundthelowerboundpN(loglogN) 3 o(1). Inarecentpreprint[46], − + HeapandLindqvistmadeapredictionbasedonrandommatrixtheorythatHelson’sconjec- tureisfalse. The preprint [26] also gave a precise answer to the question of for which m the homo- geneousDirichletpolynomials Ω(n) m,n N n−s havecomparableL4 andL2 norms. Indeed, thishappensifandonlymis,inaprec=ises≤ense,strictlysmallerthat 1loglogN. Aninteresting P 2 problem comingfrom analyticnumbertheoryand thework ofHough [54], is to extendthis resulttohighermoments. Problem 4.2. Assume k is an integer larger than 1. For which m (depending on N) will the L2k normsofm-homogeneousDirichletpolynomialsof length N becomparabletotheirL2 norms? Cancellationsin thepartialsums of theRiemann zeta function on the criticalline can be studiedthroughasimilarproblemconcerningHp norms. Problem4.3. Determinetheasymptoticbehaviorof N n 1/2 s whenN for0 n 1 − − Hp →∞ < p 1. = ≤ °P ° ° ° 6 EEROSAKSMANANDKRISTIANSEIP Aninterestingmodificationofthisproblemisthefollowing. Problem 4.4. Determine the precise asymptotic growth of k nN 1[d(n)]γn−1/2−skHp when N forp 1. = →∞ ≤ P A more general problem is to do the same for polynomials with coefficients represented by multiplicativefunctionssatisfyingappropriategrowthconditions.[22]establishedtheinequal- ity 1/2 N logp (3) µ(n) an 2[d(n)]log2−1 f Hp, Ãn 1| || | ! ≤k k X= validfor f(s) N a n s and0 p 2,whereµ(n)istheMöbiusfunction. Thisinequality, = n 1 n − < ≤ whichshouldbere=cognizedasanLp-analogueofaninequalityofHelson[50],yieldsthelower P bound N (4) n 1/2 s (logN)p/4 − − Hp ≫ n 1 ° X= ° for all 0 p . An estimate°in the oppo°site direction in the range 1 p follows by ° ° < < ∞ < < ∞ applying Helson’s theorem on the Lp boundedness of the partialsum operator [50] on suit- ablytruncatedEulerproduct. When p 1 thesamemethodyields thatan additionalfactor = loglogN appearsontheright-handsidewhen isreplacedby in(4),andthusProblems ≫ ≪ 4.3and4.4remainopenexactlyintherangep 1.SomeresultsforProblem4.4arecontained ≤ inthemanuscript[24]. A closely related and more general problem concerns the natural partial sum operator of theDirichletserieswhoseLp normcanbeestimatedbyHelson’stheorem[50]forfinitep and aresultfrom[8]forp . =∞ Problem4.5. Determinethepreciseasymptoticgrowthofthenormofthepartialsumoper- ator S : a n s N a n s when N for p 1 (or more generally, for p 1 or N ∞n 1 n − 7→ n 1 n − → ∞ = ≤ p ). = = =∞ P P Inthecasep 1,atrivialonedimensionalestimateyieldsalowerboundoforderloglogN, = whereas[24]givesanupperboundoforderlogN/loglogN,sothatpresentlythereisalarge gapbetweentheknownbounds. WefinishthissectionbyrecallingapointwiseversionoftheanalogueofHelson’sproblem onthetorus.Thus,forprimesp letχ(p):wei.i.drandomvariableswithuniformdistribution onTanddefineχ(n) ℓ χ(p )ℓl forn pk1...pkℓ. = k 1 k = 1 ℓ = Problem4.6. DeterminQethealmostsuregrowthrate(inN)ofthecharactersum N χ(n). n 1 X= Thisproblem stemsfrom Wintner,and is listedbyErdo˝s, althoughin theoriginalversion insteadχ(p):sareRademachervariables. DeepresultsontheproblemwereprovidedbyHa- lasz [43] in the 1980s, and recently Harper [44] obtained remarkable improvements for the lowerbound.Buttheoriginalproblemremainsopen. SOMEOPENQUESTIONSINANALYSISFORDIRICHLETSERIES 7 5. ESTIMATES FOR GCD SUMS AND THE RIEMANN ZETA FUNCTION Thestudyofgreatestcommondivisor(GCD)sumsoftheform N (gcd(n ,n ))2α k ℓ (5) (n n )α k,ℓ 1 k ℓ X= forα 0wasinitiatedbyErdo˝swhoinspiredGál[40]tosolveaprizeproblemoftheWiskundig > GenootschapinAmsterdamconcerningthecaseα 1. Gálprovedthatwhenα 1,theop- = = timal upper bound for (5) isCN(loglogN)2, withC an absoluteconstant independent of N and the distinct positive integers n ,...,n . The problem solved by Gál had been posed by 1 N Koksmainthe1930s,basedontheobservationthatsuchboundswouldhaveimplicationsfor theuniformdistributionofsequences(n x)mod1foralmostallx. k Using the several complex variables perspective of Bohr and seeds found in [61], Aistleit- ner, Berkes and Seip [2] proved sharp upper bounds for (5) in the range 1/2 α 1 and a < < much improved estimate for α 1/2, solving in particular a problem of Dyer and Harman = [34]. The methodof proofwas based on identifying(5) as a certain Poisson integralon D . ∞ TheacquiredboundswerealsousedtoestablishaCarleson–Hunt-typeinequalityforsystems of dilatedfunctions ofboundedvariationor belongingto Lip , a resultthatin turnsettled 1/2 two longstanding problems on the almost everywhere behavior of systems of dilated functions. The Carleson–Huntinequality and the original inequality of Gál (see (6) below) were lateroptimizedbyLewkoandRadziwiłł[60]. Additional techniques were introduced by Bondarenko and Seip [25, 26] to deal with the limitingcaseα 1/2,andfinallytherange0 α 1/2wasclarifiedin[22]. Writing = < < 1 N (gcd(n ,n ))2α Γ (N): sup k ℓ , α = N (n n )α 1 n1 n2 nNk,ℓ 1 k ℓ ≤ < <···< X= wemaysummarizethestateofaffairsasfollows: 6e2γ (6) Γ (N) loglogN 1 ∼ π2 (logN)(1 α) logΓ (N) − , 1/2 α 1 α ≍α (loglogN)α < < logNlogloglogN logΓ (N) 1/2 ≍s loglogN logΓ (N) (1 2α)logN loglogN, 0 α 1/2, α α − − ≍ < < wherein(6),γdenotesEuler’sconstant;theseestimatesremainthesameifwereplaceΓ (N) α bythepossiblylargerquantity N (gcd(n ,n ))2α Λ (N): sup c c k ℓ , α = k ℓ (n n )α 1 n1 n2 nN, c 1k,ℓ 1 k ℓ ≤ < <···< k k= X= wherethevectorc (c ,c ,...,c )consistsofnonnegativenumbersand c 2: c2 c2 c2 . = 1 2 N k k = 1+ 2+··· N 8 EEROSAKSMANANDKRISTIANSEIP Aistleitner[1] madethe importantobservationthat such estimatescan be used to obtain Ω-resultsfortheRiemannzetafunction. Indeed,usingHilberdink’sversionoftheresonance method[44],hefoundanewproofofMontgomery’sΩ-resultsforζ(α it)intherange1/2 + < α 1[64]. Inturn,BondarenkoandSeipappliedtheparticularset{n ,n ,...,n }yieldingthe 1 2 N < lowerboundforΛ (N)incombinationwiththeresonancemethodofSoundararajan[83]to 1/2 obtain(unconditionally)thefollowing: givenc 1/p2, thereexists a β, 0 β 1, such that < < < foreverysufficientlylargeT logTlogloglogT (7) sup ζ(1/2 it) exp c . t (Tβ,T)| + |≥ Ã s loglogT ! ∈ This gives an improvement by a power of logloglogT compared with previously known estimates[7,83]. p Welisttworathergeneralquestionspertainingtotheserecentdevelopments. Problem 5.1. Link the estimates for GCD sums to the function and operator theory of the spacesHp. Problem5.2. DevelopfurtherapplicationstoandlinkswiththeRiemannzetafunction. Problem 5.1 originates in the observation from [2] that GCD sums can be interpreted as Poissonintegralsonpolydiscs.TakingintoaccounttheprominentroleplayedbyPoissoninte- gralsandthePoissonkernelintheclassicalsetting(forinstanceinconnectionwithfunctions ofboundedmeanoscillation),weareledtoaskforpotentialfunctionandoperatortheoretic interpretationsorapplicationsofourestimatesforGCDsums. Finally, we would like to give an example related to the rather vague and general Prob- lem5.2. ItconcernsestimatesrelatingthesizeofthecoefficientstotheHp normofaDirich- letseries,whichcanbetracedbacktoBohr’sproblemofcomputingthemaximaldistancebe- tween theabscissasofabsoluteanduniformconvergence. BohnenblustandHille’ssolution tothisproblem[19]reliedonarevolutionarymethodofpolarizationforestimatingthesizeof thecoefficientsofhomogeneouspolynomials. TherewasarevivalofinterestinBohnenblust andHille’sworkafterthe1997paperofBoasandKhavinson[18]onso-calledBohrinequali- ties. Itwasgraduallyrecognizedthattheoriginalestimateofordermm fortheconstantC(m) intheBohnenblust–Hilleinequalitywasnotsufficientlyaccuratetoreachthedesiredlevelof precisioninvariousapplications.Basedonare-examinationoftheoriginalproof,asophisti- catedversionofHölder’sinequalityduetoBlei[17],andaKhinchin-typeinequalityofBayart [10], Defant, Frerick, Ounaïes, Ortega-Cerdà, and Seip established in [31] that C(m) grows at most exponentially in m. This was recently improved further by Bayart, Pellegrino, and Seoane-Sepúlveda[15]whowereabletoshow,bytakinganewapproachtoBlei’sinequality, thatC(m)growsatmostasexp c mlogm forsomeconstantc. ThemostimportantapplicationoftheimprovedversionoftheBohnenblust–Hilleinequal- ¡ p ¢ itywastothecomputetheSidonconstantS(N)whichisdefinedasthesupremumoftheratio between a a andsup a a 2it a Nit , withthesupremumtakenover all possib|le1c|+ho·i·c·e+s|oNf n| onzero vt∈eRctor1s+(a2,...,+a··)·i+n CNN. The following remarkably precise ¯ 1 N ¯ ¯ ¯ SOMEOPENQUESTIONSINANALYSISFORDIRICHLETSERIES 9 asymptoticresultholds[31]: 1 S(N) pNexp o(1) logNloglogN = −p2+ µ q ¶ ¡ ¢ when N . This formula has a long history and relies on the contribution of many re- → ∞ searchers, most notablyQueffélec and Konyagin [58] and de la Bretèche [28]. The proof in- volvesanunconventionalblendoftechniquesfromfunctiontheoryonpolydiscs(theBohnen- blust–Hilleinequality),analyticnumbertheory(theDickmanfunction),andprobability(the Salem–Zygmundinequality). Thereis a strikingresemblancebetween theformulafor theSidonconstantS(N) and the followingconjecturefrom[35],basedonargumentsfromrandommatrixtheory,conjectures formomentsofL-functions,andalsobyassumingarandommodelfortheprimes[35]: 1 max ζ(1/2 it) exp o(1) logTloglogT 0 t T + = −p2+ ≤ ≤ µ q ¶ ¯ ¯ ¡ ¢ whenT . Itisnatura¯ltoaskifth¯isresemblanceismorethanjustacoincidence. →∞ 6. RANDOM DIRICHLET SERIES A classical result due to Selberg states that the distributionof the Riemann zeta function onthecriticallineisasymptoticallyGaussian,aftersuitablerenormalisation.Moreprecisely, the distribution of 1loglog(T) −1/2log ζ(1/2 it) : t [0,T] tends to that of a standard 2 | + | ∈ normalvariableN (n0,1)asT .Recently,Fyodorov, KeatingoandHiarycomputedheuris- ¡ ¢ →∞ tically the covariance of the translations of the zeta function and observed that in the first approximation a logarithmic correlation structure emerges. Similar covariance structure is exhibitedby(theone-dimensional)restrictionoftheGaussianfreefield(GFF),afundamen- tal probabilistic object that figures prominently in e.g. Liouville quantum gravity, SLE and random matrix theory. Based on the classical (after Montgomery) heuristic connection between ζ(1/2 it) and random matrices, and the conjectured behaviourof randommatrices + theyproposedthefollowing Problem6.1. [39]Consider[0,T]asaprobabilityspace,withnormalisedLebesguemeasure, anddenotethecorrespondingvariablebyω [0,T]. Then,asT ,onehas ∈ →∞ 3 max log ζ(1/2 ih iω) loglogT logloglogT E, h [0,1] | + + |= −4 + ∈ wheretheerrortermE isboundedinprobabilityasT . →∞ VeryrecentlyArguin,BeliusandHarper[6]establishedtheanalogueoftheaboveconjecture foranaturalmodelthatisderivedfromtheEulerproductofthezeta-function,i.e. forpartial sumsofrandomDirichletseriesofthetype 1 X(x) cos(xlogp)cosθ sin(xlogp)sinθ , p p = pp + p X ¡ ¢ whereθ :s arei.i.d. andunifromon[0,2π]andindexedbyprimenumbers. p 10 EEROSAKSMANANDKRISTIANSEIP The GFF heuristics of the zeta-function over the critical line has been useful also in con- nectionwiththeHelsonconjecture[45]. Manyfascinatingquestionsremaintobestudiedin this general domain of probabilisticbehaviourof thezeta function and related models. For manyrandomGaussianfields(takingvaluesingeneralisedfunctions)onemayconstructthe correspondingmultiplicativeGaussianchaosmeasureseee.g. [57],[32],[9]). Naturally,after Selberg’s result one may inquireif one could produce a gaussian chaos as a suitablescaling limitoftheRiemannzetafunctiononthecriticalline.Aneasiertaskwouldbetoconsider Problem 6.2. [39] Let the field X be defined as in (8). Study the properties non-Gaussian chaos"exp(βX(x))". Someveryearlystepsinthisdirectionarecontainedin[80]. REFERENCES [1] C.Aistleitner,LowerboundsforthemaximumoftheRiemannzetafunctionalongverticallines,Math.Ann., toappear;arXiv:1409.6035. [2] C.Aistleitner,I.Berkes,andK.Seip,GCDsumsfromPoissonintegralsandsystemsofdilatedfunctions,J.Eur. Math.Soc.17(2015),1517–1546. [3] A.AlemanandJ.Cima,AnintegraloperatoronHpandHardy’sinequality,J.Anal.Math.85(2001),157–176. [4] A.Aleman, J.-F.Olsen, andE.Saksman, FouriermultipliersforHardyspacesofDirichletseries,Int.Math. Res.Not.IMRN16(2014),4368–4378. [5] A.Aleman,J.-F.Olsen,andE.Saksman,FatouandbrothersRiesztheoremsinT ,arXiv:1512.01509. ∞ [6] L.-P.Arguin,D.Belius,andA.Harper.MaximaofrandomisedRiemannzetafunctionandbranchingrandom walks,arXiv:1506.00629. [7] R.BalasubramanianandK.Ramachandra,OnthefrequencyofTitchmarsh’sphenomenonforζ(s).III,Proc. IndianAcad.Sci.Sect.A86(1977),341–351. [8] R.Balasubramanian,B.Calado,andH.Queffélec,TheBohrinequalityforordinaryDirichletseries,Studia Math.175(2006),285–304. [9] J.Barral, A.Kupiainen, M.Nikula, E.Saksman, andC.Webb,Basicpropertiesofcriticallognormalchaos, Ann.Prob.43(2015),2205–2249. [10] F.Bayart,HardyspacesofDirichletseriesandtheircompositionoperators,Monatsh.Math.136(2002),203– 236. [11] F.Bayart,CompactcompositionoperatorsonaHilbertspaceofDirichletseries,IllinoisJ.Math.47(2003), 725–743. [12] F.BayartandO.F.Brevig,Compactcompositionoperatorswithnon-linearsymbolsontheH2spaceofDirich- letseries,arXiv:1505.02944. [13] F.Bayart,A.Defant,L.Frerick,M.Maestre,andPabloSevilla-PerisMultipliersofDirichletseriesandmono- mialseriesexpansionsofholomorphicfunctionsininfinitelymanyvariables,arXiv:1405.7205. [14] F.Bayart, S.V. Konyagin, andH.Queffélec, Convergencealmosteverywhereanddivergenceeverywhereof TaylorandDirichletseries,RealAnal.Exchange29(2003/04),557–586. [15] F.Bayart,D.Pellegrino,andJ.-B.Seoane-Sepúlveda,TheBohrradiusofthen-dimensionalpolydiskisequiv- alentto (logn)/n,Adv.Math.264(2014),726–746. [16] F. Bayart, H. Queffélec, and K. Seip, Approximation numbers of composition operators on Hp spaces of p Dirichletseries,arXiv:1406.0445. [17] R.Blei,FractionalCartesianproductsofsets,Ann.Inst.Fourier(Grenoble)29(1979),79–105. [18] H.P.BoasandD.Khavinson, Bohr’spowerseriestheoreminseveralvariables,Proc.Amer.Math.Soc.,125 (1997),2975–2979. [19] H.F.BohnenblustandE.HilleOntheabsoluteconvergenceofDirichletseries,Ann.ofMath.(2)(1931),600– 622.

Some open questions in analysis for Dirichlet series

Similar Some open questions in analysis for Dirichlet series