CONVERGENCE AND MODULAR TYPE PROPERTIES OF A TWISTED RIEMANN SERIES

Description

Niveau: Supérieur, Master, Bac+5
CONVERGENCE AND MODULAR TYPE PROPERTIES OF A TWISTED RIEMANN SERIES T. RIVOAL AND J. ROQUES Abstract. We consider the series ?(?) = ∑∞ m=1 1 m2 sin(2pim 2?) cot(pim?), a twist of the famous continuous but almost nowhere differentiable sine series defined by Riemann. In a slightly different but equivalent form, this series appeared in the first author's paper [On the distribution of multiple of real numbers, Monatsh. Math 164.3 (2011), 325–360]. We pursue here the study of ?, which is almost everywhere but not everywhere convergent. We first prove that ? enjoys a modular type property, in the following sense (with ?n the n-th partial sum of ?): For all ? ? (0, 1], the sequence ?N (?) ? ??b?Nc(?1/?) has a finite simple limit ?(?) as N ? +∞. Using analytic properties of ?, we then prove that ?(?) converges if and only if ? is irrational and ∑ j log(qj+1)/qj converges (Brjuno's condition), where qj is the j-th denominator in the sequence of convergents to ?. This completes the results obtained in the above mentioned paper, where it was proved that ?(?) converges absolutely under Brjuno's condition.

• brjuno's condition


• fraction expansion


• riemann series


• j?1 ∑


• since ? ?


• see section


• sin


• functions


• convergence properties

Sujets

• Function

Informations

• Publié par: rasum
• Nombre de lectures: 15
• Langue: English

Extrait

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T. RIVOAL AND J. ROQUES

P 1 π =( ( in(2, 2 ) cot(,,)taiwtsofthe 5LefdJMf.risehert(esWisedceno) = π2 s famouscontinuousbutalmostnowheredifferentiablesineseriesdenedbyRiemann.Ina slightlydifferentbutequivalentform,thisseriesappearedintherstauthor’spaper[On the distribution of multiple of real numbers Math, Monatsh.134eW06.](201.325{31),3 pursueherethestudyof,whichisalmosteverywherebutnoteverywhereconvergent. Werstprovethatenjoysamodulartypeproperty,inthefollowingsense(with n then For-th partial sum of ): all2(0;1], the sequence  N ()  bNc ( 1=) has a nite simple limit () asN!+∞.Ungsialanicytporpitrefoseew,erpvohtne P that () converges if and only ifis irrational and α log(q α+( )q= α converges (Brjuno’s condition), whereq α is theπ-th denominator in the sequence of convergents to. This completes the results obtained in the above mentioned paper, where it was proved that () converges absolutely under Brjuno’s condition.

1.iaMfNgdaˆfA ˆ This paper deals with the series ∞ X 1 () = m + sin(2m + ) cot(m).(1.1) m=) (The summand is defined for any real numberbecause sin(2m + z) cot(mz) is an entire P function for any integerm1.ofRiwistisat)Iteissnremena hidpohantinestudyin[1R.]3amein m ∞ = n ) se m r ) i + cis (2in s s e  o m nti + It). nusuo appeared in a different form in t e onRand nowhere differentiable, except at the rational numbers of the formp/qwithp andq6]t[aarmteisDudraHeG,yrpoc(emiodthnddaboerIV]I.),5hCpatitsu;see[rver,Ita showed that these facts are simple consequences of the following modular type functional on, whereR() = P m ∞ =) m + and > equati e +im +  0: + 3 Z  ,(1. R() e i/1  0/+ R( 1/) =  6 +ie i/1 p    i 2   2 e i/ ( 1 p  R( 1/)d2) which itself can be deduced from the classical modular equation satisfied by the theta series P m2Z e im +  . The latter was used by Jaffard [7] to compute the multifractal spectrum of

Date: February 28, 2012. 2010atMcstimaheuSjbceCtalssiaction.Primary 11J70, 40A05, Secondary 11F03, 22A30. Keywordsandphrases.Twisted Riemann series, Brjuno’s condition, Modular type equations, Contin- ued fraction expansions. 0

1

Riemann series. The important information in (1.2) is that the right-hand side is much smoother (differentiable with continuous derivative on (0,+∞)than what is suggested by theleft-handside(continuousbutalmostnowheredifferentiable). Weaddressheretwoquestions: When does the series () converge? {Does the series () satisfy a modular type functional equation (like (1.2)that pro- vides some non-trivial analytic information? Although both questions might seem unrelated, it turns out that our answer to the second one is an important step to answer the first one. We shall now make a couple of comments concerning the first question. On the one hand, it is easy to prove that the series diverges for all rational number=p/qwith (p, qin this case, for any integer  Indeed,) = 1.J1, q )q ) J )J ) X X X J sin(2m + ) cot(m X 1 1 m=) m + (2ins= ) k + p/q) cot(q/pk) ) (jq+k) + + 2 j=) , qj k=) j= whencealogarithmicdivergenceof(p/qtaht]3.)nOhtoehtreahdn,itwasprovedin[1 () converges absolutely for any irrational numbersatisfyingBePhab'f BMbcL.agMaF ;baBMgMba X ∞ log(q j+) ) <∞,(1.3) q ( j j= whereq j denotes thejof the sequence of convergents to-th denominator . ( ) ) More precisely, the “absolute convergence" result proved in [13] concerns the series of general term m ) + P m k=) cos(2km) but this does not change anything since X m cm +  1 ) m + k) cos(2km) =  os(m(m m + in ) ( 1+ sm   )  (n ) is = ) sin(m + ) + = sin(2m + ) 2m c + ot(m   m + ( .4) 1 and the second fraction on the right of (1.4) is the term of an absolutely convergent series for any proof uses the fact that. Thejsin(2m + ) cot(m)j  j jm m +  j j (wherejxj denotes the distance ofx2RtoZ) and then estimates in terms of the continued fraction expansion offor the growth of the sums P N m=) m j + m +  m j j ,inspired by the estimates found j in Kruse’s paper [9] for the sums P N m ) ) j  also [14].. See = m s jm ) esriseTvedisprosultheretsehecnrc]no[n31 P n 1 (os ( c = n(n 21i )(+ n (1is) ) n 2 ) and its simple relation with ns ( (Eq.(1.4)below.Fui)gsvineybcompmoreli-id[nalectyeh41b]itndco’sepsrniioromrehtronujrB,e cated condition P α )(og+(xaq j1+ q= j q; j ) α∞httaobhtdeni1[]4snotstat.Itwa:tnelaviqueearnsioitndco q j P = ; q α ) 2q α indeed, we haveq α+( =q α ≤max(q α+( q α ≤ +( and the series α log(q α )=q α is convergent for any irrational number is well-known that only Liouville numbers may fail to satisfy Brjuno’s condition,. It which thus holds almost surely.

7iSgdP ,.

Plot of

2(( on [0 1 ,

2

In the present paper, a modular type property of () is established and used in order tounderstandmorepreciselytheconvergencepropertiesof(egeryintoran)F.Nand any real number, we denote b N ) theN-th partial sum of () : y  ( N X sin(2m + ) cot(m  () = ) . N ) m + m= We then consider

Ω N () =  N ()  ⌊ ( 1/) Nc where⌊
c observe that the limit of Ω Wedenotes the floor function. N () is. ceMbeM defined almost everywhere on (0, The first part of the paper is1] but not everywhere. devoted to the proof of the following result, which in particular shows that the limit of Ω N () exists and is defined everywhere on (0,toroirehF.]1,nemonehonchapofsuncesnsta see [1, 2, 3, 15, 18]. LWTdeTb .1abhFG;aGbFMhgd;FfFfLaTΩ N L.f . fMWcTF TMWMgΩba(0,1].fN!+∞. MbeFbiFe, X

N 1 Ω () =  sin(2m m 0 +  + ) N  ) m= jLFeFG N L.f . fMWcTF TMWMgGba[0,1].fN!+∞.aBj ;bafg.ag Gbe .TT2[0,1].aB .TTN1. In particular, the function Ω( 1 X ∞ sin(2m +  )   m=) m 0

( (

(1.5)

Mf :bhaBFB :l .a .:fbThgF

(1.6)

3

7iSgdP -.

Plot of Ω

2(( on [0.07,1]

i is defined and bounded on [0,.6(1inesrisenesiekil-nnameiRehT.1])scontinuousand behaves likelog(1/) as!0 + (see Lemma 3). These facts will be important for the proof of Theorem 2 stated below. We will provide explicit expressions for the limits G() and Ω( Graphical experiments (see) but they are not easy to study precisely. Figure 2) done with the computer algebra system PARI/GP lead us to formulate the following conjecture: 4dcjTRgheT .1TLF Gha;gMbaGMf ;bagMahbhf ba[0,1]. Fortunately, Theorem 1 provides us enough control on Ω and this conjecture is not required for the proof of our next result. LWTdeTb ..,be .al Mee.gMba.T ahW:Fe2(0,1), gLF fFeMFf();baiFeIFf MG .aB baTl MGBePhab'f;baBMgMba(1.3)LbTBf. Ia;.fFbG;baiFeIFa;F,jFL.iF ∞ X () =T()


T j ) ()Ω(T j ()(1.7) j=( jLFeFT k ()BFabgFf gLFkfigL MgFe.gF bG:l gLF G.hff W.cT() ={1/}. RFW.eR.By the classical properties of continued fraction expansions, we haveT j () q q j+) j )


T) =jq andT( j ) ( )  p j ) j  q ) j . (See Section 3.1 for some properties of j  continued fraction expansions.) Hence, since Ω has a logarithmic behavior at the origin, identity (1.7) can be viewed as the quantitative version of the first (qualitative) part of Theorem 2. Similar expansions can be found in [15, 16] for instance and implicitly in [17]. Asmentionedabove,itwasprovedin[13]that() converges absolutely if Brjuno’s condition holds. Together with Theorem 2, this leads to the following result.

4dedaaOey

TLF fFeMFf

(

;baiFeIFf MG .aB baTl MG Mg ;baiFeIFf .:fbThgFTl.

4

We observe here that  could be related with “false" Lerch’ sums and “false" theta functions (“false" means here that we replace summations overZby summations overN  intheusualdefinitionsofLerch’sumsandthetafunctions).ForLerch’sums,wereferthe reader to Mordell’s papers [11, 12] and Zwegers’ thesis [19] on mock theta functions. It would be interesting to know if an alternative expression for Ω could be deduced from this relationship which would prove Conjecture 1.

Beside Conjecture 1, let us conclude this introduction with a few other problems. For- mally,theFourierseriesexpansionof() isS()() =  )+ + + P n ∞ =) n + n + cos(2n) where X X =d +   n ifnis a square, =d + otherwise. n n 2 p p

djn )≤d≤

djn )≤d≤

This is an easy consequence of (1.4). Two problems are to determine for which’s the Fourier seriesS()() converges ( + ) and for which ones we haveS()() = (). In the spiritofDavenport-likeproblems(see[4,10]),thenaturalanswerswouldbe“ifandonly if Brjuno’s condition holds" for both problems but we don’t know what to expect here. It was proved in [13] thatS()() = () if P j ∞ =( q j+) /q j + converges, which is an almost sure condition but stronger than Brjuno’s condition. Another problem is the determination of p p the minima of  on [0,1], which seems to be at= + 2 ) and= 0  + 2 . Finally, it would be interesting to know if our method can be adapted to study the convergence of the series ∞ X sin(2m + ) cot(m) m s

m=)

for any givens2(1,iressihtrevnocseusiobvros,onasregrseidev,respges,velyecti)2(.oF for every2Rifs >2, respectively ifs≤1.)

Wewillfrequentlyworkwithanalyticfunctionshon an open subset Ω ofCdefined as the quotient of analytic functionsh=f /gon Ω. For anyz2Ω, the quotientf(z)/g(z) (which is not well defined ifg(z) = 0) will meanh(z will be implicit in the whole). This paper. We will still denote by⌊c
 ona modified floor function: [0,+∞) it coincides with the usual floor function while on (∞,0) it is set to zero. We will also often treat labels 1 given to certain quantities as mathematical expressions; for instance, if (10.) and (10.2) are such labels, we will freely write things likej(10.1)j ≤1 or (10.2) = 0 when the meaning is obvious.

+ Since

(0;1),

()convergestoalmosteverywherebyCarleson’stheorem.

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2.aRaadPVPdaPTHR, 2.1..ygYgcTYgSeYIROfOPUUg;ydddTpVeWOTgLetf:Rn {0} !C For allbe a map. N2Nand all2RnQ, we set X N f(2m + ) cot(m) Ψ N () = m=) m + . (The value of an empty sum is set to 0.) Using the classical expansion ∞ X 1 1 + + , cot(z) = z 2z n=) z n + which is valid and uniform on any compact subset ofCnZ, we get, for allM, N2Nand all2(0,+∞)nQ, + + M Ψ N () Ψ M ( 1/  1 ) = X N f(2m X f( 2m + /)   + ) 0  m=) m  m=) m 0 N X X M  f(2m + )f( 2n + /)  1 + 2 m=)n=) m   nm  +  +  n + + X N 1 X +∞ f(2m + ) + X M 1 X +∞ f( 2n + /) 2 m +  +  n +  2 mn  +  +  n + . m=) m n=M+) n=) m=N+) Iffis the restriction of an analytic function vanishing onZ, then the above equality is meaningful and valid on (0,+∞htperaitucalcrsae).erThorefine,f= sin, we obtain, for all >0, X + ⌊Nc sim + 2 Ω N () =  1 X m N ) sin(2m + ) n(/ m (12). ) 0   = m  0 )= m X X sin(2n /) 1 N ⌊Nc  sin(2m + + ) +  + 2  n  m  +  +  n + (2.2) m=)n=) m ∞ + X N 1 m X + sin(2m +  + ) )(2.3 2 m=) n=⌊Nc+) m +  +  n + 2 + X ⌊Nc 1 X +∞ si m n( +  2n +   + / n + ) .(2.4) ) n m=N+) n= InordertoproveTheorem1,itissucienttoshowthatthethreesequences(2.,(2.3) 2) and (2.4) converge asNtends to +∞and that their moduli are bounded by an absolute constant for all2[0,1] andN will 1. Thisbe proved in Sections 2.2, 2.3 and 2.4 respectively.