Arithmetical functions: an introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties

392 Pages·1994·12.51 MB·English

by  Wolfgang Schwarz

CAMBRIDGE UNIVERSITY PRESS

131 Algebreaxitcr,e mamle trciocm binatoMr-iMc.Ds E,Z A,P .F RANKL I.GR.O SENBERG( eds)
132 Whitehegraodu op&fsf inigtreo upRsO,B ERTO LIVER &
133 Lineaalrg ebrmaoincoi dMsO,H AN S.P liTCHA
134 Numbetrh eoarnydd ynamiscyaslt emMs.,D ODSON J.V ICKER(Se ds)
135 Operaatlogre barnads applic1a,tD i.Eo VnA sN,S M&.T AKESAK(Ie ds)
136 Operaatlogre barnadas p plica2t,iD o.nE sV,A NS& M.T AKESAK(Ie ds)
137 AnalyasitUs r banIa,,E .B ERKSON,T .P ECK&, J.U HL( eds)
138 AnalyastiUs r banIaI,,E .B ERKSON,T .P ECK&, J.U HL( eds)
139 Advanciesnh omototphye orSy.,S ALAMdN,B. S &T EER W.S liTHERLAN(De ds)
140 Geometraiscpe ctosfB anacshp aceEs.,M P.E INADOaRn &dA .R ODES( eds)
141 Surveycso mbinat1o9r8i9cJ,s. S IEMONS (ed)
142 The geoinm etroyfj ebtu ndles, D.J. SAUNDERS
143 Thee rgodUi1ce oorfyd iscrgertoeu pPsE,T ERJ .N ICHOLLS
144 Introdutcotu inoinf osrpma ceIs.,M J.A MES
145 Homologiqcuaels tiionln osc aallg ebrJaA,NR .S TROOKER
146 CohenMacamuodlualye osv eCro hen-MacaruilnagyYs .,Y OSHINO
147 Continuaonudds i scrmeotdeu leSs.,H M.O HAMED B.JM.U LLER
148 Helicaensdv ectbourn dleAs.,N R.U DAKOV &
149 Solitnoonnsl inear evolutioinn veeeqtrsa usc/aea t titoenMrs.i A nBgL,O WITZ P.C LARKSON
150 Geometorfyl owdimensmiaonniafl&o 1 l,dS s.D ONALDSON C.BT.H OM&A S( eds)
151 Geometorfly o wdimensmiaonniaflo 2l,dS s.D ONALDSON& C.BT.H OMAS (eds)
152 Oligomorppehrmiuct atgiroonu pPs.,C AMERON &
153 I.functaindoa nrsi thmeJt.Ci OcA,T ES M.JT.A YLOR( eds)
154 Numbetrh eoarnydc ryptograJp.Lh OyX&,T ON( ed)
155 Classifictahteiooronif pe osl arizveadr ietTiAeKsA,O F UJITA
156 Twistionrms a themaatnidcp sh ysicTs.,N B.A ILEY R.JB.A STON (eds)
157 Analyptrioc- gpr oupJs.,D D.I XONM,. P.DF.U SA&U TOYA, .M ANN D.S EGAL
158 GeometorfyB anacshp acePs.,F .MX0.L LER W.S CHACHERMAY&E R( eds)
159 GroupSstA ndrew1s9 8v9o lum1e, C.M.C AM&P BELL E.FR.O BERTSON( eds)
160 GroupSstA ndrew1s9 8v9o lum2e, C .M.C AMPBELL& E.FR.O BERTSON( eds)
161 Lectuorneb sl octkh eorByU,R KHARDK 0LSHAMME&R
162 Harmonainca lyasnidrs e presentthaetoifrooyngr r ouapcst ionngh omogenetoruese s,
A.F IGATALAMANCA C.N EBBIA
163 Topiicnsv arieotfig erso& ur pe presentaSt.iMoV.nO sV,S I
164 Quasisymmdeetsriigcn Ms.,S S.H RIKANDE S.SS.A NE
165 Groupcso,m binatorgiecosm etrMy.,W .L IE&B ECK J.S AXL (eds)
166 Surveiyncs o mbina&t o1r9i9c1As,., D K.E EDWELL( e&d )
167 Stochaasntailcy sMi.sT,B. A RLOW N.HB.I NGHAM( eds)
168 Representoafta ilognesb rHa.sT ,A C&HI KWA A S.B RENNER( eds)
169 Booleafnu nctcioomnp lexiMtSy.,P ATERSO&N( ed)
170 Manifowlidtssh i ngulaarnidUt 1iAeed sa msNoviskpoevc tsreaqlu encBe.B, O TVINNIK
171 SquareAs.,R R.A JWADE
172 Algebrvaairci etGiEeOsR,G ER .K EMPF
173 Discrgertoeu apnsd g eometrWy.,JH .A RVEY C.M ACLACHLAN( eds)
174 Lectuornem se chaniJc.sE,M. A RSDEN &
175 Adamsm emorisaylm posiounma lgebrtaoipco l1og,Ny . R AY G.W ALKER( eds)
176 Adamsm emorisaylm posiounam l gebrtaoipco l2og,Ny . R AY & G.W ALKER( eds)
177 Applicatoifco antse goirnic eosm putsecri encMe.,P F.O URMA&N ,P .TJ.O HNSTONE,
A.M.P ITTS( eds)
178 L&o wer aKn dL theorAy.,R A NICKI
179 Complex projectiveG .E gLeLoImNeGtSrRyU,CD .,P ESKINEG,.S ACCHIERO
S.AS.T R0MME( eds)
180 L&e ctuornee sr goditch eoarnydP esiUn1 eoornyc ompamcatn ifolMd.sP ,O LLICOTT
181 Geometgrriocu tph eoIr,yG .AN.I BLO M A.R OLLER( eds)
182 Geometgrriocu tph eoryG .AN.I BL&O M.A.R OLLER( eds)
183 Shintzaentifa u ncti1oAn1.Ys ,U, K IE &
184 Arithmetfiucnaclt ioWn.sS ,C HWARZ J.S PILKER
185 Representaotfis oonlsv agbrloeu ps,M &A NZ T.RW.O LF
186 Complexkintoyt:cs o,l ourainndgc so0 u.n tinDg.&,J .WAE.L SH
187 Surveiyncs o mbinato1r9i9c3sK,,. W ALKER( ed)
189 Localplrye sentaanbdal cec essciabtleeg orJi.Ae DsA,M EK J.R OSICKY
190 Polynomiinavla rioafnf itsn igtreo upDs,J .B ENSON &
191 Finigteeo metarnydc ombinatoFr.iD cEs C,L ERCK
192 Symplectic geDo.Sm AeLtArMyO,N ( ed) eta /
197 Two-dimenshioomnoatlo apnydc ombinatgorroiutaphl e orCy.,H OG-ANGELONI
W.M ETZLER A.JS.I ERADSK(Ie ds)
198 Thea lgebrc&ahia cr acteroifgz eaotimoent4 r-imca nifoJlAd.Hs ,I LLMAN
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Properties
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Firpsutb lished 1994
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ISBN0 5214 27285
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Cotnetns
Preafce xi
Acknowledgments
XV
Noattion xvii
ChaptIe r Toolfsrm o Number Theory 1
1.P1a.r tSiuamlm ation 2
12..A rithmeFtuiniccaotlnC so,vn oliuotnM,ob iIunsv erFsoiromnu 4l a
1.P3e.r ioFduinccn tsiE,ov eFnu nicotnRsa,m anuSjuamns 15
14..T he Tu-rKaunbiIlnieuqlsuyi a t 19
15..G eneraFtuinnicgot nDsi,r icShelreite s 25
1.S6o.m Ree sulotnPs r imNeu mber.s . . 31
I.C 7h.a racLt-eFrusno,c ntsPi,r imiensA rithmPertoiegcsr sio3ns5
18..E xercises 39
Photopghrsa 43
Chapt1e1r Mean VaThleuoer emasn dM ultipliFcuantcnits,vi eoI 4 5
II.M1o.t ivat.i o.n . . . . . . . . .4 6. . . . . . . . .
II.E2l.e nmtearMye na-ValTuhee ore(mWsl ntAnxeerr,) 49
II.E3s.t imaftoeSrsu msov eMru ltciaptliiFvuen cti(oRnasn kin's
Trick.) . . . . . . . . . 5.6 . . . . . . . . . . .
II.W4i.rn sgi'Mse an-VaTlhueeo rfeomSr u msov eNro n-Nteigvae
MultitpilvFieuc nacti.o n.s . . . . . 6.5 . . . . . . .
IIS.T.h eT heoroefmG .H alaosnzM ean-VaolfuC eosm plex-
ValuMeudl tiicpaltFiuvnec ti.on.s. 76
II.T6h.eT heoroefmD aboussDie laannodgn te h Fe ourier-Coef-
ficieonfMt usl ptliitciavFeu nctio.n s. . . 7.8 .
I7I..A pplicoaftti hoeDn a bouDseslia-nTghee orteoma Problem
ofU nifoDrims tribution 81
I.I8T.h eT heoroefmS affaanrdDi a bousIs.i, 82
II.9. DaEbloeumsaesrniyPt 'rosof tohfPe r imNeu mbeTrh eore8m5
II.M1o0h.na NairE'lse mteanrMye thoidnP rimNeu mbeTrho ery9 1
-vi-i
II.E1x1e.r cises 93
Chapter RelatAerdi thmeFtuincicaotln s 97
Ill
IIII.n1t.r oduMcotitivoant,i on 98
II.I2M.a iRne sults 101
IIIL.e3m.mt aaP,r oof of 2T.h3e orem 104
III.A4p.p lica.ti.o.ns. . ... . . . . 110
II5I.O. n Tah eoroefmL .L ucht . . . 115
IIIT.h6eT. h eoroefmS affaanrdDi a bousIsIi. , 117
II.I7A.p plictaotA ilomnto -sPeriFoudnicct ions 118
III.E8x.e rci.s.es. . . . . . . ... . . . . .12.1. .
ChaptIeVr UniforAmllmyo st PeArriiotdhimce tical
Functi.o ns 123
IV1..E veann dP erioAdriict hmeFtuinccatli ons 124
IV.S2i.m pPlreo perties 133
IV.L3i.m Ditir situbtions 139
IV.4G.e lfaTnhde'yos:rM aximal SIpdaecaels 142
u
IV..4T .hAem aximiadle sapla c!:2e o fJJ 142
u
IV.4T.hBem. a ximiadle al t:s:Dp oaf.Vc e 147
IV5..A pplicoaftT iioent zEex'tse nsTihoeno rem 155
IV.I6n.tr eagtioofUn n iforAmllmyo st-FEuvnecnt ions 165
IV7..E xercises 162
ChaptVe r RamanujEaxnpn asioonfFs u nctiionn s 165
2u .
V.1I.n troduc.t i.o n . . . . . . . . 1.6 6. . . . . .
V.2E.q uivaloefTn hceeo re1m.s12 .,13 .,14 ., 5 168
V.3S.o mLee mmata 171
V4..P roooffT heor1e5.m 175
V.5.P roooffL emmsa 3.a4n d3 5. 178
V.6E.x ercises 184
ChaptVeIr AlmoPsetr-ioadnidAc l most-AErvietnh metical
Functions 185
VI.1B.e sicoNvoircmSh,p acesA lmoofP setor diiFcu nction1s8 6
VI.2S.o mPer operotfSi peasc esq -Aolfm ost-PFeurnicotdii1oc9n 7s
-vii-i