Some open questions in analysis for Dirichlet series

0.16 MB·

by  Eero Saksman

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 vari-
ables. 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 func-
tions. 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 be-
tween 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].
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