irrational Many values of the Riemann zeta function at odd integers are C.R. Acad. Sci. Paris, Ser.I

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Number theory

Many values of the Riemann zeta function at odd integers are irrational

Beaucoup de valeurs aux entiers impairs de la fonction zêta de Riemann sont irrationnelles

Stéphane Fischler

^a

, Johannes Sprang

^b

, Wadim Zudilin

^c

^,

^d

aLaboratoiredemathématiquesd’Orsay,UniversitéParis-Sud,CNRS,UniversitéParis-Saclay,91405Orsay,France bFakultätfürMathematik,UniversitätRegensburg,93053Regensburg,Germany

cDepartmentofMathematics,IMAPP,RadboudUniversity,POBox9010,6500GLNijmegen,theNetherlands dSchoolofMathematicalandPhysicalSciences,TheUniversityofNewcastle,Callaghan,NSW2308,Australia

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received29March2018

Acceptedafterrevision17May2018 Availableonline28May2018 PresentedbyJean-PierreSerre

Inthisnote,weannouncethefollowingresult:atleast2^{(1−ε)log logslogs} valuesoftheRiemann zetafunctionatoddintegersbetween3andsareirrational,where

ε

isanypositivereal numberandsislargeenoughintermsof

ε

.Thisimprovesonthelowerbound₁₊^1−ε_{log 2}logs that followsfromthe Ball–Rivoaltheorem. Wegive the mainideas oftheproof, which isbasedonaneliminationprocessbetweenseverallinearformsinoddzetavalueswith relatedcoefficients.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Dans cette note, on annonce le résultat suivant : au moins 2^{(1−ε)log loglogss} valeurs de la fonction zêtade Riemannauxentiers impairscompris entre3and s sontirrationnelles, où

ε

estunréelstrictementpositifetsunentierimpairsuffisammentgrandenfonction de

ε

.Ceciaméliorelaborne₁₊^1−ε_{log 2}logsquidécouleduthéorèmedeBall–Rivoal.Ondonne lesidéesprincipalesdelapreuve,quiestfondéesurunprocédéd’éliminationentredes formeslinéairesenlesvaleursdezêtaauxentiersimpairsdontlescoefficientssontreliés.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Versionfrançaiseabrégée

Danscettenote, onannoncelerésultat suivant(voirlaversion anglaisepourunbrefhistorique).Onnedonnequeles idéesprincipalesdelapreuve(voir[5] pourlapreuvecomplète).

E-mailaddresses:stephane.fi[email protected](S. Fischler),[email protected](J. Sprang),[email protected](W. Zudilin).

https://doi.org/10.1016/j.crma.2018.05.007

1631-073X/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Théorème1.Soient

ε >

0ets≥^{3unentierimpairsuffisammentgrand(enfonctionde}

ε

).Alors,parmilesnombres

ζ (

3

), ζ (

5

), ζ (

7

), . . . , ζ (

s

),

aumoins

2^{(1−ε)log loglogss} sontirrationnels.

Soient

ε >

0 et sun entierimpair suffisamment grand(en fonctionde

ε

). Onnote D leproduit de touslesnombres premiersinférieursouégauxà

(

1−²

ε )

logs,detellesortequ’ona D≤^{s1−ε.Ensuivant}essentiellement[10],onconsidère, pourtoutentiern,lafractionrationnelle

Rn

(

t

) =

^D3Dnn

!

^s+1−3D

_3Dn

j=0

(

t

−

ⁿ

+

_D^j

)

n

j=0

(

t

+

^j

)

^s+1

et,pourtout j∈ {¹

, . . . ,

D}^,onpose r_n,j

=

∞ m=1

Rn

m

+

^j

D

.

RappelonsquelafonctionzêtadeHurwitzestdéfiniepar

ζ (

i

, α )

=_∞

n=0 1

(n+α)ⁱ pour

α >

0 eti≥^2.EndéveloppantRn

(

t

)

enélémentssimples,ondémontre,commedans[10],que

r_n,j

= ρ

0,j

+

3≤ⁱ≤^s iimpair

ρ

i

ζ

i

,

^j D

.

Enoutre,ennotantdnlepluspetitcommunmultipledesnpremiersentiers,onmontrequed_n^s+₊¹₁estunmultiplecommun desdénominateursdesnombresrationnels

ρ

0,jet

ρ

i.

Lepointcrucial,commedans[10],estl’identitésuivante,valablepourtoutdiviseurddeD etpourtoutentieri≥2 :

d

j=1

ζ

i

,

^j

D d

D

=

d

j=1

ζ

i

,

^j d

=

^∞

n=⁰

d

j=1

dⁱ

(

dn

+

^j

)

ⁱ

=

^di

ζ (

i

).

Eneffet,cetterelationmontrequ’enposantrn,d=d j=1r_n,jD

d,ona :

rn,d

=

d

j=1

ρ

_0,jD

d

+

3≤i≤s iimpair

ρ

idⁱ

ζ (

i

).

Notons D ^{l’ensemble des diviseursde D. Pour toutd}∈D ^{et pourtout entiern, ona donc une}combinaison linéaire r_n,d de1,

ζ (

3

)

,

ζ (

5

)

,. . . ,

ζ (

s

)

.Pour toutentieri impaircompris entre3ets,lecoefficientde

ζ (

i

)

est

ρ

idⁱ (oùlenombre rationnel

ρ

i dépendimplicitementden,maispasded).Sidesentiersw_d,d∈D^,vérifient

d∈Dw_ddⁱ=^{0 pouruncertain} i,alorslacombinaisonlinéaire

r_n

=

d∈D

w_dr_n,d (0.1)

ne faitplus apparaître

ζ (

i

)

.Le point central dece procédé d’élimination(mis au point par letroisièmeauteur [12] pour D=^{2,etgénéraliséparledeuxièmeauteur[10])estquelecoefficientdidépendded,maispasden:onpeutchoisirles} entiers wd indépendammentden.

Pour démontrer le théorème 1, on suppose que, parmi

ζ (

3

)

,

ζ (

5

)

, . . . ,

ζ (

s

)

, le nombre d’irrationnels est inférieur à 2⁽¹⁻³ε)_{log log}^logs_s

;ilestalorsinférieurouégalà

δ

−^1,où

δ

=^CardD^{estlenombredediviseursdeD.Ilexistedoncdesindices} impairsi1

<

i2

< . . . <

iδ−¹ comprisentre3etstelsquetoutevaleurirrationnelle

ζ (

i

)

,aveci impaircompris entre3ets, soitl’unedes

ζ (

i_j

)

.Onpeutchoisirdesentiersrelatifs w_d nontousnuls(pourd∈D^)telsque

d∈D

w_dd^ij

=

0 pour toutj

∈ {

1

, . . . , δ −

1

}.

(0.2)

Larelation(0.1) définitalorsunesuite

(

rn

)

n≥¹denombresréelsquisontdescombinaisonslinéairesàcoefficientsrationnels de 1 etdes

ζ (

i

)

pour i impair,3≤ⁱ≤^{s.La propriété cruciale estque les}

ζ (

i_j

)

nefigurent plus dansces combinaisons linéaires,sibienquetousles

ζ (

i

)

quiyapparaissentavecuncoefficientnonnulsont,parhypothèse,desnombresrationnels.

En multipliant par un dénominateur commun A de ces nombres rationnels, et aussi par d_n^s+₊¹₁ (quiest un dénominateur commun des coefficients

ρ

i et

ρ

0,j), on obtientune suite d’entiers relatifs.Une étude asymptotique des suites

(

rn,j

)

n≥¹ montrequecettesuited’entiersrelatifstendvers0,doncestidentiquementnulleàpartird’uncertain rang ;précisément, ona,quandn→ ∞^:

Ad_n^s+₊¹₁

rn

=

d∈D

w_dd

+

^o

(

1

)

Ad_n^s+₊¹₁rn,1

,

avec 0

<

lim

n→∞

Ad^s_n⁺₊¹₁rn,1

1/n

<

1

où o

(

1

)

est une suite qui tend vers 0 quand n tend vers l’infini. Cela impose

d∈Dw_dd=^{0. Autrement dit, tout fa-} mille

(

w_d

)

_d∈D d’entiers relatifs vérifiant (0.2) devrait vérifier aussi

d∈Dw_dd=^{0. Ce n’est pas le cas, car la matrice} [^dij]d∈D,0≤^j≤δ−¹ (dans laquelleon posei₀=^{1) est}inversible : sondéterminant estleproduitd’un déterminantde Van- dermondeetd’unpolynômedeSchur,quiestunpolynômeàcoefficientsentiersnaturelsévaluéenlafamilledesdiviseurs de D.

1. Introduction

When s≥^{2 is an eveninteger, thevalue}

ζ (

s

)

oftheRiemannzetafunction isa non-zerorationalmultipleofπ^{s and,} therefore,atranscendentalnumber.Ontheotherhand,veryfewresultsare knownonthezetavalues

ζ (

s

)

whens≥^{3 is} odd,thoughweexpectthemalltobetranscendental.

It was onlyin 1978whenApéry astonished themathematicscommunity by hisproof [1] ofthe irrationalityof

ζ (

3

)

, withthenextbreakthroughinthisdirectiontakenin2000byBallandRivoal[2,8],whoprovedthefollowingtheorem.

Theorem1(Ball–Rivoal).Let

ε >

0,ands≥^{3beanoddintegersufficientlylargewithrespectto}

ε

.Then dim_QSpan_Q

(

1

, ζ (

3

), ζ (

5

), ζ (

7

), . . . , ζ (

s

)) ≥

¹

− ε

1

+

^{log 2logs}

.

Inspiteof severalrefinements[2,11,4] for smalls,thelower bound inTheorem 1hasnever beenimprovedforlarge valuesofs.TheproofofTheorem1appliesNesterenko’slinearindependencecriterion[7] tocertainlinearcombinationsof oddzetavalues.Toimproveonthisboundusingthesamestrategy,onehastofindlinearcombinationsthatareconsiderably smaller,withnottoolargecoefficients — thiscomesouttobearatherdifficulttask.

Thesituationhasdrasticallychangedwhenthethirdauthorintroduced[12] anewmethod,whichhasbeengeneralized bythesecondauthorin[10].Foragiveninteger D

>

1 andacertainrationalfunction R

(

t

)

,theseries

d

j=1

∞ t=¹ R

t

+

^j

d

,

whered

|

D

,

givesQ^-linearcombinationsof1

,

d³

ζ (

3

),

d⁵

ζ (

5

), . . . ,

d^s

ζ (

s

)

withcoefficientsindependentofd(exceptthatof1).Thismakes itpossibletoeliminate fromtheentirecollectionoftheselinearcombinationsessentiallyasmanyoddzetavaluesasthe numberofdivisorsofD.Forapplicationsofthisidea,see[12,6,10,9].

In thisnote, we sketch the proof of the following result, buildingupon the approach in[12] and[10]. We refer the interestedreadertothefullversion[5] ofthepaperfordetails.

Theorem2.Let

ε >

0,ands≥^{3beanoddintegersufficientlylargewithrespectto}

ε

.Thenamongthenumbers

ζ (

3

), ζ (

5

), ζ (

7

), . . . , ζ (

s

),

atleast

2^{(1−ε)log loglogss} areirrational.

Incomparison,Theorem 1givesonly ¹⁻ε

1+^{log 2}logsirrational oddzetavalues,buttheyarelinearlyindependentoverthe rationals,whereasTheorem2endsuponlywiththeirirrationality.

OurproofofTheorem2followstheabove-mentionedstrategyfrom[12] and[10].Themainnewingredient,compared to the proof in [10], is taking D large (about s¹⁻ε) and equal to the product ofthe first prime numbers (the so-called primorial) — suchanumberhasasymptoticallythelargestpossiblenumberofdivisorswithrespecttoitssize.

2. SmalllinearformsinvaluesoftheHurwitzzetafunction

Let sand D be positiveintegers suchthat sisodd ands≥^{3D.Letnbe apositive integer,such that Dniseven. We} considerthefollowingrationalfunction:

R_n

(

t

) =

^D3Dnn

!

^s+1−3D

3Dn

j=⁰

(

t

−

ⁿ

+

_D^j

)

_n

j=⁰

(

t

+

^j

)

^s+1

which,ofcourse,dependsalsoonsandD,andfor j∈ {1

, . . . ,

D}welet rn,j

=

∞ m=1

Rn

m

+

^j

D

.

We write d_n for lcm

(

1

,

2

, . . . ,

n

)

.Expanding R_n

(

t

)

into partial fractionsyields thefollowing Q^-linearcombinations of valuesoftheHurwitzzetafunction.

Lemma1.Foreach j∈ {¹

, . . . ,

D}^,wehave

r_n,j

= ρ

0,j

+

3≤ⁱ≤^s i odd

ρ

i

ζ

i

,

^j D

with

d_n^s+1−i

ρ

i

∈ Z

^{for i}

∈ {

³

,

5

, . . . ,

s

}

^{and ds}_n⁺₊¹₁

ρ

0,j

∈ Z

^{for any j}

∈ {

¹

, . . . ,

D

}.

Wepointoutthatthecoefficient

ρ

iof

ζ (

i

,

_D^j

)

inthelinearformrn,j doesnotdependon j.Theexpansionofrn,j isvery classical; onlyoddzetavaluesappearbecauseofthesymmetryphenomenonfrom[2].Theproofthatd^s_n⁺¹⁻ⁱ

ρ

i∈Z^forodd ifollowsthatof[3,Lemma 4.5].Thelastassertion,namelyd_n^s+₊¹₁

ρ

0,j∈Z^{,isprovedasin[10,}Lemma 1.4].

An important feature ofthe linear formsrn,j is that they are simultaneouslyvery small: even when multipliedby a commondenominatoroftherationalcoefficients,theystilltendto0.

Lemma2.Assumethat_Dlog^s _D largerthansomeeffectivelycomputableabsoluteconstant.Thenwehave

nlim→∞r_n^1/_,ⁿ_j

=

g

(

x0

) <

3^−(s+1) and lim

n→∞

r_n,j

rn,j

=

1 for any j

,

j

∈ {

1

, . . . ,

D

},

where

g

(

x

) =

^D3D

(

x

+

3

)

^3D

(

x

+

1

)

^s+1

(

x

+

2

)

^2(s+1)

andx0istheuniquepositiverootofthepolynomial

(

X+³

)

^D

(

X+¹

)

^s+1−^XD

(

X+²

)

^s+1.

The followingelementarylemmaaboutthenon-vanishingofgeneralizedVandermondedeterminantsisusedtogetrid ofunwantedirrationaloddzetavalues(in§3below).

Lemma3.Fort≥^1,letx1

, . . . ,

xtbepairwisedistinctpositiverealnumbersand

α

1

, . . . , α

tpairwisedistinctnon-negativeintegers.

Thenthematrix[^xα_jⁱ]1≤i,j≤thasnon-zerodeterminant.

3. Eliminationofoddzetavalues

Take0

< ε <

¹₃,andletsbeoddandsufficientlylargewithrespectto

ε

.WeletD betheproductofallprimeslessthan orequalto

(

1−²

ε )

logs,sothat D≤^{s1−ε.Noticethat D has}

δ

=^{2π((1−2ε)logs)divisors,withlog}

δ

≥

(

1−³

ε )(

log 2

)

_{log log}^logs_s. Assume that thenumberofirrational oddzetavaluesbetween

ζ (

3

)

and

ζ (

s

)

islessthanorequal to

δ

−^1.Let3=ⁱ1

<

i2

< . . . <

i_δ−1≤^{sbe oddintegers such that if}

ζ (

i

)

∈

/

Q ^{andi isodd, 3}≤ⁱ≤^{s, then i}=ⁱj for some j.Moreover, we let i0=^{1,andconsidertheset}DofalldivisorsofD,sothat CardD=

δ

.Lemma3showsthat thematrix[^dij]d∈D,0≤^j≤δ−¹ is invertible.Therefore,thereexistintegers w_d∈Z^,whered∈D^,suchthat

d∈D

w_dd^ij

=

0 for any j

∈ {

1

, . . . , δ −

1

}

and

d∈D

w_dd

=

0

.

(3.1)

Thecrucialpoint(asin[10,§3])isthatforanyd∈Dandanyi≥^2,

d

j=¹

ζ

i

,

^j

D d

D

=

d

j=¹

ζ

i

,

^j d

=

^∞

n=⁰

d

j=¹ dⁱ

(

dn

+

j

)

ⁱ

=

^di

ζ (

i

).

Lemma1impliesthatthequantitiesrn,d=d j=¹r_n,jD

d arelinearformsintheoddzetavalues:

rn,d

=

d

j=1

ρ

_0,jD

d

+

3≤i≤s iodd

ρ

idⁱ

ζ (

i

),

(3.2)

whileLemma2leadstothefollowingasymptotics:

r_n,d

= (

d

+

^o

(

1

))

r_n,1 with lim

n→∞r¹_n,^/₁ⁿ

=

^g

(

x₀

).

(3.3)

Weshallusenowtheintegers wd toeliminate

δ

−^{1 oddzetavalues,includingallirrationalones, asin[12] and[10].}

Forthat,introduce^rn=

d∈Dw_dr_n,d.Eqns. (3.1)–(3.3) implythat

rn

=

d∈D wd

d

j=1

ρ

_0,jD d

+

i∈I

ρ

i d∈D

wddⁱ

ζ (

i

) =

d∈D

wdd

+

o

(

1

)

rn,1

,

where I= {³

,

5

,

7

, . . . ,

s}\ {ⁱ1

, . . . ,

i_δ−1}^.In particular,no irrationalzetavalue

ζ (

i

)

,where3≤ⁱ≤^{s,appearsin thislinear} combination, and limn→∞|rn|^1/n=^g

(

x0

) <

3^−(s+1) since

d∈Dw_dd=^{0. Denoting by A a common} denominator ofthe (rational)numbers

ζ (

i

)

,i∈^{I,wededucefromLemma1that Ads}n⁺+¹¹rnisaninteger.Nowtheprimenumbertheoremimplies thatd_n^1/₊ⁿ₁→^easn→ ∞^{,hencefromLemma2weconcludethatthesequenceofintegerssatisfies}

0

<

lim

n→∞

|

^Ads_n⁺₊¹₁

rn

|

^1/n

=

^es+1g

(

x0

) <

_e 3

s+1

<

1

.

ThiscontradictionimpliesthetruthofTheorem2.

Acknowledgements

WethankMichelWaldschmidtforhisadvice,andOleWarnaarforhiscommentsonanearlierdraftofthepaper.

References

[1]R.Apéry,Irrationalitédeζ (2)etζ (3),in:JournéesArithmétiques,Luminy,1978,in:Astérisque,vol. 61,1979,pp. 11–13.

[2]K.Ball,T.Rivoal,Irrationalitéd’uneinfinitédevaleursdelafonctionzêtaauxentiersimpairs,Invent.Math.146 (1)(2001)193–207.

[3]S.Fischler,Shidlovsky’smultiplicityestimateandirrationalityofzetavalues,preprint,arXiv:1609.09770 [math.NT],J.Aust.Math.Soc.(2018),inpress.

[4]S.Fischler,W.Zudilin,ArefinementofNesterenko’slinearindependencecriterionwithapplicationstozetavalues,Math.Ann.347(2010)739–763.

[5]S.Fischler,J.Sprang,W.Zudilin,Manyoddzetavaluesareirrational,preprint,arXiv:1803.08905 [math.NT],2018.

[6]C.Krattenthaler,W.Zudilin,Hypergeometryinspiredbyirrationalityquestions,preprint,arXiv:1802.08856 [math.NT],2018.

[7]Y.Nesterenko,Onthelinearindependenceofnumbers,Vestn.Mosk.Univ.,Ser.Filos.(Mosc.Univ.Math.Bull.)40 (1)(1985)46–49(69–74).

[8]T.Rivoal,LafonctionzêtadeRiemannprenduneinfinitédevaleursirrationnellesauxentiersimpairs,C.R.Acad.Sci.Paris,Ser.I331 (4)(2000) 267–270.

[9]T.Rivoal,W.Zudilin,Anoteonoddzetavalues,preprint,arXiv:1803.03160 [math.NT],2018.

[10]J.Sprang,Infinitelymanyoddzetavaluesareirrational.Byelementarymeans,preprint,arXiv:1802.09410 [math.NT],2018.

[11]W.Zudilin,IrrationalityofvaluesoftheRiemannzetafunction,Izv.Akad.NaukSSSR,Ser.Mat.(Izv.Math.)66 (3)(2002)49–102(489–542).

[12]W.Zudilin,Oneoftheoddzetavaluesfromζ (5)toζ (25)isirrational.Byelementarymeans,SIGMA14 (028)(2018)(8pages).