BRIEF COMMUNICATIONS Diophantine Problems for q-Zeta Values

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1、MathematicalNotes,vol.72,no.6,2002,pp.858–862.TranslatedfromMatematicheskieZametki,vol.72,no.6,2002,pp.936–940.OriginalRussianTextCopyrightc2002byV.V.Zudilin.BRIEFCOMMUNICATIONSDiophantineProblemsforq-ZetaValuesV.V.ZudilinReceivedJune3,2002Keywords:irrationalitymeasure,z

2、etavalue,basichypergeometricseries,Eisensteinseries,hypergeometrictransformation.1.INTRODUCTIONAsusual,quantitiesdependingonanumberqandbecomingclassicalobjectsasq→1(atleastformally)areregardedasq-analogsorq-extensions.Apossiblewaytoq-extendthevaluesoftheRiemannzetafuncti

3、onreadsasfollows(hereq∈C,

4、q

5、<1):X∞X∞νk−1qνX∞qνρ(qν)nkζq(k)=σk−1(n)q=1−qν=(1−qν)k,k=1,2,...,(1)n=1ν=1ν=1Pwhereσ(n)=dk−1isthesumofpowersofthedivisorsandthepolynomialsρ(x)∈Z[x]k−1d

6、nkcanbedeterminedrecursivelybytheformulasρ=1andρ=(1+(k−1)x)ρ+x(1−x)ρ′1k+1kkfork=1,2,...(see[1

7、,Part8,Chap.1,Sec.8,Problem75]forthecasek=2).Thenthelimitrelationslim(1−q)kζ(k)=ρ(1)·ζ(k)=(k−1)!·ζ(k),k=2,3,...,(2)qkq→1

8、q

9、<1hold;theequalityρk(1)=(k−1)!isprovedin[2,formula(7)].Theabovedefinedq-zetavalues(1)presentseveralnewinterestingproblemsinthetheoryofdiophantineappr

10、oximationsandtranscendentalnumbers;theseproblemsareextensionsofthecorrespondingproblemsforordinaryzetavaluesandwestatesomeoftheminSec.3ofthisnote.Ournearestaimistodemonstratehowsomerecentcontributionstothearithmeticstudyofthenumbersζ(k),k=2,3,...,successfullyworkforq-zet

11、avalues.Namely,wemeanthehypergeometricconstructionoflinearforms(proposedintheworksofE.M.Nikishin[3],L.A.Gutnik[4],Yu.V.Nesterenko[5])andthearithmeticmethod(duetoG.V.Chudnovsky[6],E.A.Rukhadze[7],M.Hata[8])accompaniedbythegroup-structurescheme(duetoG.RhinandC.Viola[9],[10

12、]).Thenextsectioncontainsnewirrationalitymeasuresofthenumbersζ(1)andζ(2)forq−1=p∈Z{0,±1},andourstartingqqpointisthefollowingtableillustratingaconnectionofsomeobjectsandtheirq-extensions(here⌊·⌋denotestheintegralpartofanumberandthenotation‘l.c.m.’meanstheleastcommonmulti

13、ple).Wereferthereadertothebook[11]andthepapers[12]–[14],wheresomemotivationsandjustificationsarepresente