Handbook of number theory II

635 Pages·2004·2.15 MB·English

by  Jozsef Sandor, Borislav Crstici

HANDBOOK
OF NUMBER THEORY II
by
J.Sa´ndor
Babes¸-BolyaiUniversityofCluj
DepartmentofMathematicsandComputerScience
Cluj-Napoca,Romania
and
B.Crstici
formerlytheTechnicalUniversityofTimis¸oara
Timis¸oaraRomania
KLUWER ACADEMIC PUBLISHERS
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Contents
PREFACE 7
BASICSYMBOLS 9
BASICNOTATIONS 10
1 PERFECTNUMBERS:OLDANDNEWISSUES;
PERSPECTIVES 15
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Somehistoricalfacts . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Evenperfectnumbers . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Oddperfectnumbers . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Perfect,multiperfectandmultiplyperfectnumbers . . . . . . . . . 32
1.6 Quasiperfect,almostperfect,andpseudoperfect
numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.7 Superperfectandrelatednumbers . . . . . . . . . . . . . . . . . . 38
1.8 Pseudoperfect,weirdandharmonicnumbers . . . . . . . . . . . . . 42
1.9 Unitary,bi-unitary,infinitary-perfectandrelatednumbers . . . . . . 45
1.10 Hyperperfect,exponentiallyperfect,integer-perfect
andγ-perfectnumbers . . . . . . . . . . . . . . . . . . . . . . . . 50
1.11 Multiplicativelyperfectnumbers . . . . . . . . . . . . . . . . . . . 55
1.12 Practicalnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.13 Amicablenumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.14 Sociablenumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References 77
2 GENERALIZATIONSANDEXTENSIONSOFTHE
MO¨BIUSFUNCTION 99
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
1
CONTENTS
2.2 Mo¨biusfunctionsgeneratedbyarithmeticalproducts
(orconvolutions) . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
1 Mo¨biusfunctionsdefinedbyDirichletproducts . . . . . . . 106
2 UnitaryMo¨biusfunctions . . . . . . . . . . . . . . . . . . 110
3 Bi-unitaryMo¨biusfunction . . . . . . . . . . . . . . . . . 111
4 Mo¨biusfunctionsgeneratedbyregularconvolutions . . . . 112
5 K-convolutionsandMo¨biusfunctions. B convolution. . . . 114
6 ExponentialMo¨biusfunctions . . . . . . . . . . . . . . . . 117
7 l.c.m.-product(vonSterneck-Lehmer) . . . . . . . . . . . . 119
8 Golomb-GuerinconvolutionandMo¨biusfunction . . . . . . 121
9 max-product(Lehmer-Buschman) . . . . . . . . . . . . . . 122
10 InfinitaryconvolutionandMo¨biusfunction . . . . . . . . . 124
11 Mo¨biusfunctionofgeneralized(Beurling)integers . . . . . 124
12 Lucas-Carlitz(l-c)productandMo¨biusfunctions . . . . . . 125
13 Matrix-generatedconvolution . . . . . . . . . . . . . . . . 127
2.3 Mo¨biusfunctiongeneralizationsbyothernumbertheoretical
considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
1 Apostol’sMo¨biusfunctionoforderk . . . . . . . . . . . . 129
2 Sastry’sMo¨biusfunction . . . . . . . . . . . . . . . . . . . 130
3 Mo¨biusfunctionsofHanumanthachariand
Subrahmanyasastri . . . . . . . . . . . . . . . . . . . . . . 132
4 Cohen’sMo¨biusfunctionsandtotients . . . . . . . . . . . . 134
5 Klee’sMo¨biusfunctionandtotient . . . . . . . . . . . . . . 135
6 Mo¨biusfunctionsofSubbaraoandHarris;Tanaka;
andVenkataramanandSivaramakrishnan . . . . . . . . . . 136
7 Mo¨biusfunctionsascoefficientsofthecyclotomic
polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.4 Mo¨biusfunctionsofposetsandlattices . . . . . . . . . . . . . . . . 139
1 Introduction,basicresults . . . . . . . . . . . . . . . . . . 139
2 Factorableincidencefunctions,applications . . . . . . . . . 143
3 Inversiontheoremsandapplications . . . . . . . . . . . . . 145
4 Mo¨biusfunctionsonEulerianposets . . . . . . . . . . . . . 146
5 Miscellaneousresults . . . . . . . . . . . . . . . . . . . . . 148
2.5 Mo¨biusfunctionsofarithmeticalsemigroups,freegroups,
finitegroups,algebraicnumberfields,andtracemonoids . . . . . . 148
1 Mo¨biusfunctionsofarithmeticalsemigroups . . . . . . . . 148
2 FeeabeliangroupsandMo¨biusfunctions . . . . . . . . . . 151
3 Mo¨biusfunctionsoffinitegroups . . . . . . . . . . . . . . 154
2
CONTENTS
4 Mo¨biusfunctionsofalgebraicnumberand
function-fields . . . . . . . . . . . . . . . . . . . . . . . . 159
5 TracemonoidsandMo¨biusfunctions . . . . . . . . . . . . 161
References 163
3 THEMANYFACETSOFEULER’STOTIENT 179
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
1 Theinfinitudeofprimes . . . . . . . . . . . . . . . . . . . 180
2 Exactformulaeforprimesintermsofϕ . . . . . . . . . . . 180
3 Infiniteseriesandproductsinvolving
ϕ,Pillai’s(Cesa`ro’s)arithmeticfunctions . . . . . . . . . . 181
4 Enumerationproblemsoncongruences,directed
graphs,magicsquares . . . . . . . . . . . . . . . . . . . . 183
5 Fouriercoefficientsofevenfunctions(modn) . . . . . . . . 184
6 Algebraicindependenceofarithmeticfunctions . . . . . . . 185
7 Algebraicandanalyticapplicationoftotients . . . . . . . . 186
8 ϕ-convergenceofSchoenberg . . . . . . . . . . . . . . . . 187
3.2 CongruencepropertiesofEuler’stotientandrelatedfunctions. . . . 188
1 Euler’sdivisibilitytheorem . . . . . . . . . . . . . . . . . . 188
2 Carmichael’sfunction,maximalgeneralizationof
Fermat’stheorem . . . . . . . . . . . . . . . . . . . . . . . 189
3 Gauss’divisibilitytheorem . . . . . . . . . . . . . . . . . . 191
4 Minimal,normal,andaverageorderofCarmichael’s
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5 Divisibilitypropertiesofiterationofϕ . . . . . . . . . . . . 195
6 Congruencepropertiesofϕ andrelatedfunctions . . . . . . 201
7 Euler’stotientinresidueclasses . . . . . . . . . . . . . . . 204
8 Primetotatives . . . . . . . . . . . . . . . . . . . . . . . . 206
9 Thedualofϕ,noncototients . . . . . . . . . . . . . . . . . 208
10 Eulerminimumfunction . . . . . . . . . . . . . . . . . . . 210
11 Lehmer’sconjecture,generalizationsandextensions . . . . 212
3.3 EquationsinvolvingEuler’sandrelatedtotients . . . . . . . . . . . 216
1 Equationsoftypeϕ(x +k) = ϕ(x) . . . . . . . . . . . . . 216
2 ϕ(x +k) = 2ϕ(x +k) = ϕ(x)+ϕ(k)andrelated
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
3 Equationϕ(x) = k,Carmichael’sconjecture . . . . . . . . 225
4 Equationsinvolvingϕ andotherarithmeticfunctions . . . . 230
5 Thecompositionofϕ andotherarithmeticfunctions . . . . 234
6 Perfecttotientnumbersandrelatedresults . . . . . . . . . . 240
3
CONTENTS
3.4 Thetotatives(ortotitives)ofanumber . . . . . . . . . . . . . . . . 242
1 Historicalnotes,congruences . . . . . . . . . . . . . . . . 242
2 Thedistributionoftotatives . . . . . . . . . . . . . . . . . 246
3 Addingtotatives . . . . . . . . . . . . . . . . . . . . . . . 248
4 Addingunits (mod n) . . . . . . . . . . . . . . . . . . . . 249
5 Distributionofinverses (mod n) . . . . . . . . . . . . . . 250
3.5 Cyclotomicpolynomials . . . . . . . . . . . . . . . . . . . . . . . 251
1 Introduction,irreducibilityresults . . . . . . . . . . . . . . 251
2 Divisibilityproperties . . . . . . . . . . . . . . . . . . . . 253
3 Thecoefficientsofcyclotomicpolynomials . . . . . . . . . 256
4 Miscellaneousresults . . . . . . . . . . . . . . . . . . . . . 261
3.6 Matricesanddeterminantsconnectedwithϕ . . . . . . . . . . . . . 263
1 Smith’sdeterminant . . . . . . . . . . . . . . . . . . . . . 263
2 Poset-theoreticgeneralizations . . . . . . . . . . . . . . . . 266
3 Factor-closed,gcd-closed,lcm-closedsets,and
relateddeterminants . . . . . . . . . . . . . . . . . . . . . 270
4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 273
3.7 GeneralizationsandextensionsofEuler’stotient . . . . . . . . . . . 275
1 Jordan,Jordan-Nagell,vonSterneck,Cohen-totients . . . . 275
2 Schemmel,Schemmel-Nagell,Lucas-totients . . . . . . . . 276
3 Ramanujan’ssum . . . . . . . . . . . . . . . . . . . . . . . 277
4 Klee’stotient . . . . . . . . . . . . . . . . . . . . . . . . . 278
5 Nagell’s,Adler’s,Stevens’,KesavaMenon’stotients . . . . 278
6 Unitary,semi-unitary,bi-unitarytotients . . . . . . . . . . . 281
7 Alladi’stotient . . . . . . . . . . . . . . . . . . . . . . . . 282
8 Legendre’stotient. . . . . . . . . . . . . . . . . . . . . . . 283
9 Eulertotientsofmeetsemilatticesandfinitefields . . . . . 285
10 Nonunitary,infinitary,exponential-totients . . . . . . . . . 287
11 Thacker’s,Leudesdorf’s,Lehmer’s,Golubev’stotients.
Squaretotient,core-reducedtotient,M-voidtotient,
additivetotient . . . . . . . . . . . . . . . . . . . . . . . . 289
12 Eulertotientsofarithmeticalsemigroups,finitegroups,
algebraicnumberfields,semigroups,finitecommutative
rings,finiteDedekinddomains . . . . . . . . . . . . . . . . 292
References 295
4 SPECIALARITHMETICFUNCTIONSCONNECTEDWITH
THEDIVISORS,ORWITHTHEDIGITSOFANUMBER 329
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
4
CONTENTS
4.2 Specialarithmeticfunctionsconnectedwiththedivisors
ofanumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
1 Maximumandminimumexponents . . . . . . . . . . . . . 330
2 Theproductofexponents. . . . . . . . . . . . . . . . . . . 332
3 Arithmeticfunctionsconnectedwiththeprimepower
factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
4 Otherfunctions;thederivedsequenceofanumber . . . . . 336
5 Theconsecutiveprimedivisorsofanumber . . . . . . . . . 337
6 Theconsecutivedivisorsofaninteger . . . . . . . . . . . . 342
7 Functionallimittheoremsfortheconsecutivedivisors . . . 343
8 Miscellaneousarithmeticfunctionsconnectedwith
divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
9 Arithmeticfunctionsofconsecutivedivisors . . . . . . . . . 349
10 Hooley’s(cid:3)function . . . . . . . . . . . . . . . . . . . . . 360
11 ExtensionsoftheErdo¨sconjecture(theorem) . . . . . . . . 363
12 Thedivisorsinresidueclassesandinintervals . . . . . . . 363
13 Divisordensityanddistribution (mod 1)ondivisors . . . . 366
14 Thefractalstructureofdivisors . . . . . . . . . . . . . . . 367
15 Thedivisorgraphs . . . . . . . . . . . . . . . . . . . . . . 369
4.3 Arithmeticfunctionsassociatedtothedigitsofanumber . . . . . . 371
1 Theaverageorderofthesum-of-digitsfunction . . . . . . . 371
2 Boundsonthesum-of-digitsfunction . . . . . . . . . . . . 376
3 Thesumofdigitsofprimes . . . . . . . . . . . . . . . . . 379
4 Nivennumbers . . . . . . . . . . . . . . . . . . . . . . . . 381
5 Smithnumbers . . . . . . . . . . . . . . . . . . . . . . . . 383
6 Selfnumbers . . . . . . . . . . . . . . . . . . . . . . . . . 384
7 Thesum-of-digitsfunctioninresidueclasses . . . . . . . . 387
8 Thue-MorseandRudin-Shapirosequences . . . . . . . . . 390
9 q-additiveandq-multiplicativefunctions . . . . . . . . . . 401
10 Uniform-andwell-distributionsofαs (n) . . . . . . . . . 410
q
11 TheG-arydigitalexpansionofanumber . . . . . . . . . . 414
12 Thesum-of-digitsfunctionfornegativeintegerbases . . . . 417
13 Thesum-of-digitsfunctioninalgebraicnumberfields . . . . 418
14 Thesymmetricsigneddigitalexpansion . . . . . . . . . . . 421
15 Infinitesumsandproductsinvolvingthesum-of-digits
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
16 Miscellaneousresultsondigitalexpansions . . . . . . . . . 427
References 433
5
CONTENTS
5 STIRLING,BELL,BERNOULLI,EULERAND
EULERIANNUMBERS 459
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
5.2 StirlingandBellnumbers . . . . . . . . . . . . . . . . . . . . . . . 459
1 Stirlingnumbersofbothkinds,Lahnumbers . . . . . . . . 459
2 IdentitiesforStirlingnumbers . . . . . . . . . . . . . . . . 464
3 GeneralizedStirlingnumbers . . . . . . . . . . . . . . . . 469
4 CongruencesforStirlingandBellnumbers . . . . . . . . . 488
5 Diophantineresults . . . . . . . . . . . . . . . . . . . . . . 507
6 Inequalitiesandestimates . . . . . . . . . . . . . . . . . . 508
5.3 BernoulliandEulernumbers . . . . . . . . . . . . . . . . . . . . . 525
1 Definitions,basicpropertiesofBernoullinumbers
andpolynomials . . . . . . . . . . . . . . . . . . . . . . . 525
2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
3 CongruencesforBernoullinumbersandpolynomials.
Euleriannumbersandpolynomials . . . . . . . . . . . . . . 539
4 Estimatesandinequalities . . . . . . . . . . . . . . . . . . 574
References 585
Index 619
6
Preface
Theaimofthisbookistosystematizeandsurveyinaneasilyaccessiblemanner
themostimportantresultsfromsomepartsofNumberTheory,whichareconnected
withmanyotherfieldsofMathematicsorScience.Eachchaptercanbeviewedasan
encyclopediaoftheconsideredfield,withmanyfacetsandinterconnectionswithvir-
tuallyalmostallmajortopicsasDiscretemathematics,Combinatorialtheory,Numer-
ical analysis, Finite difference calculus, Probability theory; and such classical fields
of mathematics as Algebra, Geometry, and Mathematical analysis. Some aspects of
Chapter1and3onPerfectnumbersandEuler’stotient,havebeenconsideredalsoin
our former volume ”Handbook of Number Theory” (Kluwer Academic Publishers,
1995),incooperationwiththelateProfessorD.S.Mitrinovic´ofBelgradeUniversity,
aswellasProfessorB.Crstici,formerlyofTimis¸oaraTechnicalUniversity.However,
therewereincludedmainlyestimatesandinequalities,whichareindeedveryuseful,
but many important relations (e.g. congruences) were left out, giving a panoramic
viewofmanyotherpartsofNumberTheory.
Thisvolumeaimsalsotocomplementtheseissues,andalsotobringtotheatten-
tion of the readers (specialists or not) the hidden beauty of many theories outside a
givenfieldofinterest.
Thisbookfocusestoo,astheformervolume,onsomeimportantarithmeticfunc-
tions of Number Theory and Discrete mathematics, such as Euler’s totient ϕ(n) and
its many generalizations; the sum of divisors function σ(n) with the many old and
new issues on Perfect numbers; the Mo¨bius function, along with its generalizations
and extensions, in connection with many applications; the arithmetic functions re-
latedtothedivisors,consecutivedivisors,orthedigitsofanumber.Thelastchapter
showsperhapsmoststrikinglythecross-fertilizationofNumbertheorywithCombi-
natorics,Numericalmathematics,orProbabilitytheory.
The style of presentation of the material differs from that of our former volume,
since we have opted here for a more flexible, conversational, survey-type method.
Eachchapterisconcludedwithadetailedandup-to-datelistofReferences,whileat
theendofthebookonecanfindanextensiveSubjectindex.
7
PREFACE
We have used a wealth of literature, consisting of books, monographs, journals,
separates, reviews from Mathematical Reviews and from Zentralblatt fu¨r Mathe-
matik, etc. This volume was not possible to elaborate without the kind support of
many people. The author is indebted to scientists all over the world, for providing
him along the years reprints of their papers, books, letters, or personal communi-
cations. Special thanks are due to Professors A. Adelberg, G. Andrews, T. Agoh,
R.Askey,H.Alzer,J.-P.Allouche,K.Atanassov,E.Bach,A.Blass,W.Borho,P.B.
Borwein,D.W.Boyd,D.Berend,R.G.Buschman,A.Balog,A.Baker,B.C.Berndt,
R. de la Brete`che, B. C. Carlson, C. Cooper, G. L. Cohen, M. Deaconescu, R. Dus-
saud, M. Drmota, J. De´sarme´nien, K. Dilcher, P. Erdo¨s, P. D. T. A. Elliott, M. Eie,
S. Finch, K. Ford, J. B. Friedlander, J. Fehe´r, A. A. Gioia, A. Grytczuk, K. Gyo¨ry,
J.Galambos,J.M.DeKoninck,P.J.Grabner,H.W.Gould,E.-U.Gekeler,P.Hagis,
Jr., D. R. Heath-Brown, H. Harborth, P. Haukkanen, A. Hildebrand, A. Hoit, F. T.
Howard,L.Habsieger,J.J.Holt,A.Ivic´,H.Iwata,K.-H.Indlekofer,F.Halter-Koch,
H.-J.Kanold,M.Kishore,I.Ka´tai,P.A.Kemp,E.Kra¨tzel,T.Kim,G.O.H.Katona,
P. Leroux, A. Laforgia, A. T. Lundell, F. Luca, D. H. Lehmer, A. Makowski, M. R.
Murthy,V.K.Murthy,P.Moree,H.Maier,E.Manstavicˇius,N.S.Mendelsohn,J.-L.
Nicolas,E.Neuman,W.G.Novak,H.Niederhausen,C.Pomerance,Sˇ.Porubsky´,L.
Panaitopol,J.E.Pecˇaric´,Zs.Pa´les,A.Peretti,H.J.J.teRiele,B.Rizzi,D.Redmond,
N. Robbins, P. Ribenboim, I. Z. Ruzsa, H. N. Shapiro, M. V. Subbarao, A. Sa´rko¨zy,
A.Schinzel,R.Sivaramakrishnan,J.Sura´nyi,T.Sˇala´t,J.O.Shallit,K.B.Stolarsky,
B. E. Sagan, I. Sh. Slavutskii, F. Schipp, V. E. S. Szabo´, L. To´th, G. Tenenbaum, R.
F. Tichy, J. M. Thuswaldner, Gh. Toader, R. Tijdeman, N. M. Temme, H. Tsumura,
R.Wiegandt,S.S.Wagstaff,Jr.,Ch.Wall,B.Wegner,M.Wo´jtowicz.
The author wishes to express his gratitude also to a number of organizations
whomhereceivedadviceandsupportinthepreparationofthismaterial.Thesearethe
Mathematics Department of the Babes¸-Bolyai University, the Alfred Re´nyi Institute
of Mathematics (Budapest), the Domus Hungarica Foundation of Hungary, and the
Sapientia Foundation of Cluj, Romania. The gratefulness of the author is addressed
tothestaffofKluwerAcademicPublishers,especiallytoMr.MarliesVlot,Ms.Lynn
BrandonandMs.LiesbethMolfortheirsupportwhiletypesettingthemanuscript.
The camera-ready manuscript for the present book was prepared by
Mrs.GeorgetaBonda(Cluj)towhomtheauthorexpresseshisgratitude.
Theauthor
8