Values of the Euler phi function not divisible by

arXiv:math/0611509v1[math.NT]16Nov2006ValuesoftheEulerphifunctionnotdivisiblebyaprescribedoddprimePieterMoreeAbstractLetϕdenoteEuler’sphifunction.ForafixedoddprimeqwegiveanasymptoticseriesexpansioninthesenseofPoincar´eforthenumberEq(x)ofn≤xsuchthatq∤ϕ(n).TherebyweimproveonarecenttheorembyB.K.SpearmanandK.S.Williams[Ark.Mat.44(2006),166–181].Furthermoreweresolve,undertheGeneralizedRiemannHypothesis,whichoftwoapproximationstoEq(x)isasymptoticallysuperiorusingrecentresultsofY.IharaontheEuler-Kroneckerconstantofanumberfield.1IntroductionLetϕdenoteEuler’sphifunction.ForafixedoddprimeqwesetEq={n|q∤ϕ(n)}andletEq(x)denotetheassociatedcountingfunction.(IfAisanysetofintegers,thenbytheassociatedcountingfunctionA(x)wedenotethecardinalityoftheelementsainAsuchthata≤x.)SpearmanandWilliams[16]provedthat,asxtendstoinfinity,Eq(x)=e(q)xlog1/(q−1)x1+Oǫ(1log1−ǫx),(1)withe(q)=(q+1)(q−1)q−2q−1Γ(1q−1)sin(πq−1)2q−32(q−1)q3(q−2)2(q−1)π32(h(q)R(q)C(q))1q−1,(2)whereh(q)denotestheclassnumberofthecyclotomicfieldK(q):=Q(ζq)andR(q)itsregulator.SpearmanandWilliamsgavearatherinvolveddescriptionofC(q),seeSection3,butwewillshowthatactuallyC(q)=C(q,1),whereforRe(s)1/2,C(q,s)=Yp6=qfp≥21−1psfpq−1fp,(3)wherethesumisoverallprimesp6=qsuchthatfp,thesmallestintegerk≥1suchthatpk≡1(modq),satisfiesfp≥2.OnehasC(3)=Qp≡2(mod3)(1−1/p2)forexample(thisisLemma3.1of[16]).MathematicsSubjectClassification(2000).11N37,11Y601ThegoalofthisnoteistopointoutthatthetheoryofFrobenianfunctionsallowsonetoproveanestimateforEq(x)whichismuchmoreprecisethan(1),namely(5).Moreover,wewillshowthatmakinguseoftheEulerproductfortheDedekindzetafunctionofacyclotomicnumberfield,cf.(9),leadstoasimplificationoftheargumentsofSpearmanandWilliams.ItallowsoneforexampletoinferthatC(q)=C(q,1)andtogiveaveryshortproofoftheestimate(32).ThetheoryofFrobenianfunctionswasinitiatedbyLandau[5](and,indepen-dently,butonlyheuristically,byRamanujan[8]),continuedbyBernays(oflaterfameinlogic)inhisPhDthesisandmuchlaterbySerre[14]andbroughtinitspresentstatebyOdoni,seee.g.[10].ForourpurposesTheorem1,moreorlessimplicitintheworkofLandaualready,willdo.Beforestatingit,wefirstdefinewhataFrobeniansetofprimesis.AsetofprimesPiscalledFrobeniusofdensityδ,ifthereexistsafiniteGaloisextensionK/QandasubsetHofG:=Gal(K/Q)suchthat•Hisstableunderconjugation;•|H|/|G|=δ;•foreveryprimep,withatmostfinitelymanyexceptions,onehaspinPifσp(K/Q)isinH,whereσp(K/Q)denotestheFrobeniusmapofpinG(definedmoduloconjugationincasepdoesnotdividethediscriminantofK).Theorem1[14].LetEbeasetofintegersandE′itscomplementinthesetofnaturalnumbers.SupposethatE′ismultiplicative,thatisifaandbarecoprimepositiveintegers,thenab∈E′⇐⇒{a∈E′orb∈E′}.Puth(s)=Pn∈E′n−s.LetPbethesetofprimesthatareinE.SupposethatPisFrobenianofdensityδ,with0δ1.Thenh(s)/shasanexpansionaroundthepoints=1oftheformh(s)s=1(s−1)1−δ(c0+c1(s−1)+···+ck(s−1)k+···).Furthermore,foreveryintegerk≥2wehaveE′(x)=xlogδxe0+e1logx+···+eklogkx+O(1logk+1x),withej=cj/Γ(1−j−δ).InourproblemathanditturnsoutthatPisthesetofprimesp≡1(modq).ButthisispreciselythesetofprimespthatsplitcompletelyinK(q)andthusζK(q)(s),theDedekindzetafunctionofK(q),comesintoplay.Weputα(q):=Ress=1ζK(q)(s).ThereaderunfamiliarwiththismaterialisreferredtoSection2.Theorem2Letqbeanoddprime.Puthq(s)=(1−q−2s)ζ(s)(C(q,s)(1−q−s)ζK(q)(s))1q−1.(4)2Thenhq(s)/shasanexpansionaroundthepoints=1oftheformhq(s)s=1(s−1)(q−2)/(q−1)c0(q)+c1(q)(s−1)+···+ck(q)(s−1)k+···,Foranyk≥2wehaveEq(x)=xlog1/(q−1)xe0(q)+e1(q)logx+···+ek(q)logkx+O(1logk+1x),(5)whereej(q)=cj(q)/Γ(q−2q−1−j).Inparticular,e0(q)=(1−1q2)Γ(q−2q−1)(C(q)(1−1q)α(q))1q−1.OnusingthatΓ(1q−1)Γ(q−2q−1)=πsinπq−1andformula(8),itisseenaftersomeeasycomputationthate(q)=e0(q).Thustheestimate(1),thatisthetheoremofSpearmanandWilliams,isaweakerformofTheorem2.Inthesecondpartofthepaperwedealwiththeproblemofwhetherthee(q)xlog1/(q−1)x−naive−ore(q)Zx2dtlog1/(q−1)t−Ramanujantype−approximationyields-asymptotically-abetterapproximationtoEq(x).ForeveryoddprimeqthiscanbedecidedusingTheorem5.UsingrecentresultsofIhara[3],whichassumetheGeneralizedRiemannHypothesis(GRH)tobetrue,wewillestablishthefollowingtheorem.Theorem3(GRH).Letqbeanoddprime.Forq≤67theRamanujantypeap-proximationisasymptoticallybetterthanthenaiveapproximationforEq(x),forallremainingprimesthenaiveapproximationisasymptoticallybetter.2PreliminariesForageneralnumberfieldKwehave,forRe(s)1,ζK(s):=Xa1Nas=Yp11−Np−s.Theletterawillbeusedtodenoteanon-zeroidealinOK,theringofintegersofK,andpwillbeusedtodenoteanon-zeroprimeidealinOK.ThisfunctionistheDedekindzetafunctionofK.ItisknownthatthesumandproductconvergeforRe(s)1,thatζK(s)canbeanalyticallycontinuedtoaneighborhoodof1(infact,tothewholecomplexplane),andthatats=1ithasasimplepole.3LetαKdenotetheresidueofthepoleats=1.ItisknownthatαK=2r1(2π)r2hKRKwKp|dK|,(6)wherer1isthenumberofrealinfiniteprimes,r2isthenumberofcomplexinfiniteprimes,h(K)istheclassnumberofK,R(K)istheregulatorofK,w(K)isthenumberofrootsofunityinK,andD(K)isthediscriminantofK.Arounds=1wehavetheLaurentexpansionζK(s)=αKs−1+γK+γ1(K)(s−1)+γ2(K)(s−1)2+···(7)Theconstantsγj(Q)arekn