Table Of ContentHandbook of Number Theory I
Handbook of Number
Theory I
by
József Sándor
Babes-Bolyai University of Cluj,
Cluj-Napoca,
Romania
Dragoslav S. Mitrinović
formerly of the University of Belgrade,
Servia
and
Borislav Crstici
formerly of the Technical University of Timisoara,
Romania
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-4215-9 (HB)
ISBN-13 978-1-4020-4215-7 (HB)
ISBN-10 1-4020-3658-2 (e-book)
ISBN-13 978-1-4020-3658-3 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
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1st ed. 1995. 2nd printing
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Printed in the Netherlands.
TABLE OF CONTENTS
PREFACE....................................................................................... xxv
BASICSYMBOLS............................................................................ 1
BASICNOTATIONS.......................................................................... 2
ChapterI
EULER’Sϕ-FUNCTION..................................................................... 9
§I.1 Elementaryinequalitiesfor(cid:1) .............................................. 9
§I.2 Inequalitiesfor(cid:1)(mn) ....................................................... 9
§I.3 Relationsconnecting(cid:1),(cid:2),d ............................................... 10
§I.4 Inequalitiesfor J ,(cid:2) ,(cid:3) ................................................... 11
k k k
§I.5 Unitaryanaloguesof J ,(cid:2) ,d ............................................. 12
k k
§I.6 Compositionof(cid:1),(cid:2),(cid:3) ..................................................... 13
§I.7 Compositionof(cid:2),(cid:1) ......................................................... 13
§I.8 Onthefunctionn/(cid:1)(n) ...................................................... 14
§I.9 Minimumof(cid:1)(n)/nforconsecutivevaluesofn ....................... 15
§I.10 On(cid:1)(n+1)/(cid:1)(n) ........................................................... 16
§I.11 On((cid:1)(n+1),(cid:1)(n)) ......................................................... 18
§I.12 On(n,(cid:1)(n))................................................................... 18
§I.13 Thedifferenceofconsecutivetotients ................................... 19
§I.14 Nonmonotonicityof(cid:1).(Ameasure) .................................... 19
§I.15 Nonmonotonicityof J ..................................................... 20
k
§I.16 Numberofsolutionsof(cid:1)(x)=n! ........................................ 20
§I.17 Numberofsolutionsof(cid:1)(x)=m ........................................ 21
§I.18 Numberofvaluesof(cid:1)lessthanorequaltox ......................... 22
§I.19 Oncompositenwith(cid:1)(n)|(n−1)(Lehmer’sconjecture) ........... 23
§I.20 N(cid:1)umberofcompositen ≤ x with(cid:1)(n)|(n−1) ........................ 24
§I.21 (cid:1)(n)........................................................................ 24
n≤x (cid:2) (cid:3)
(cid:1)
k
§I.22 f ·(cid:1)(k) ............................................................ 25
n
k≤n(cid:1)
3
§I.23 On (cid:1)(n)− x2 ........................................................ 25
(cid:4)2
(cid:1)n≤x
§I.24 On (cid:1)(n)/n ................................................................ 27
n≤x
vi TableofContents
(cid:1)
§I.25 On J (n)−xk+1/(k+1)(cid:5)(k+1) ................................... 28
k
n≤x
§I.26 Anexpansionof J .......................................................... 29
(cid:4) k
§I.27 On 1/(cid:1)(n)andrelatedquestions .................................. 29
(cid:4) n≤x
§I.28 (cid:1)(p−1)for pprime ............................................... 30
p(cid:4)≤x
§I.29 On (cid:1)(f(n)), f apolynomial ....................................... 31
(cid:4) n≤x (cid:4)
§I.30 (cid:1)∗(n), (cid:1)(n)(cid:1)(n+k)andrelatedresults ................. 31
n≤x n≤x
§I.31 AsymptoticformulaeforgeneralizedEulerfunctions ................ 32
(cid:4)
§I.32 On(cid:1)(x,n)= 1andonJacobstahl’sarithmetic
m≤x,(m,n)=1
function ........................................................................ 33
§I.33 Ontheiterationof(cid:1) ........................................................ 34
§I.34 Iteratesof(cid:1)andtheorderof(cid:1)(k)(n)/(cid:1)(k+1)(n) ........................ 35
§I.35 Normalorderof(cid:6)((cid:1)(n)) ................................................... 36
ChapterII
THEARITHMETICALFUNCTIONd(n),ITSGENERALIZATIONSANDITS
ANALOGUES.................................................................................. 39
§II.1 Thedivisorfunctionsatconsecutiveintegers .......................... 39
§II.2 Ond(n+i )>···>d(n+i ) .......................................... 40
1 r
§II.3 Relationsconnectingd,(cid:6),(cid:2),d ......................................... 41
k
§II.4 Ond(mn) ..................................................................... 42
§II.5 Aninequalityford (n) ..................................................... 42
k
§II.6 Majorizationforlogd(n)/log2 .......................................... 42
§II.7 maxd(n)andmax(d(n),d(n+1))andgeneralizations .............. 44
n≤x n≤x
§II.8 Highlycomposite,superiorhighlycomposite,andlargely
compositenumbers.......................................................... 45
§II.9 Congruen(cid:1)cepropertyofd(n) .............................................. 47
§II.10 (cid:3)(x)= d(n)−xlogx −(2(cid:7) −1)x ............................... 47
(cid:1) n≤x
§II.11 d(p−1), pprime ..................................................... 49
p≤x (cid:1)
§II.12 (cid:3)k(x)= dk(n)−x · Pk−1(logx),k ≥2 ........................... 51
(cid:1) n≤x
§II.13 d2(n) ..................................................................... 55
k
n≤x(cid:1)
§II.14 On (g∗d )(n) .......................................................... 55
k
n≤x
§II.15 (cid:3) (x).......................................................................... 56
3
§II.16 The(cid:1)divisorprobleminarithmeticprogressions ...................... 57
§II.17 On 1/d (n) .............................................................. 59
k
n≤x
TableofContents vii
§II.18 Averageorderofd (n)overintegersfreeoflargeprime
k
factors ......................................................................... 60
§II.19 Onasumond andLegendre’ssymbol ................................ 60
k
§II.20 Asu(cid:1)mondk,d and(cid:6) ...................................................... 61
§II.21 On d(n)·d(n+N)andrelatedproblems ......................... 61
(cid:1)n≤x
§II.22 On d (n)·d(n+1)andrelatedquestions ......................... 63
k
n≤x
§II.23 Iterationofd ................................................................. 65
§II.24 Ond(cid:1)(f(n))andd(d(f((cid:1)n))), f apolynomial .......................... 66
§II.25 On d(n2+a)and d(m2+n2) ................................ 67
(cid:1)n≤x m,n≤x
§II.26 d(|f(r,s)|), f(x,y)abinarycubicform .................... 68
|f(r,s)|≤N
§II.27 Weig(cid:1)hteddivisorproblem ................................................. 68
§II.28 On d(n−kt) .......................................................... 69
k<n1/t
§II.29 Divisorsumsonsquarefreeorsquarefullintegers .................... 69
§II.30 Exponentialdivisors ....................................................... 71
§II.31 Bi-unitarydivisors .......................................................... 72
§II.32 (cid:1)Sumsoverd(n)·(cid:6)(n),d(n)/(cid:6)(n),(cid:6)(d(n)),(cid:4)(d(n)) ................ 72
§II.33 d(a(n)),a(n)thenumberofabeliangroupswithn
n≤x
elements ...................................................................... 73
§II.34 d(n)inshortintervals ...................................................... 73
§II.35 Numberofdistinctvaluesofd(n)for1≤n ≤ x ..................... 74
§II.36 Onthedistributionfunctionofd(n) ..................................... 74
§II.37 On(nd(n),(cid:2)(n))=1...................................................... 75
§II.38 Averagevalueforthenumberofdivisorsofsumsa+b ............ 75
ChapterIII
SUM-OF-DIVISORSFUNCTION,GENERALIZATIONS,ANALOGUES;
PERFECTNUMBERSANDRELATEDPROBLEMS................................ 77
§III.1 Elementaryinequalitieson(cid:2)(n)and(cid:2)(n)/n ......................... 77
§III.2 On(cid:2)(n)/nloglogn ........................................................ 79
§III.3 On(cid:2) (n)/nk ................................................................. 80
(cid:4) k (cid:4) (cid:4)
§III.4 (cid:2)(n), (cid:2)(n), (cid:2)(n) ................................... 81
n≤x n≤x,p|n n≤x,(n,k)=1
(cid:2)(n)
§III.5 Sumsover ............................................................ 82
n
§III.6 Sumsover(cid:2)k(n) ............................................................ 83
§III.7 Onsumsover(cid:2)−(cid:8)(f(n)), f apolynomial(0<(cid:8)<1) ............. 84
viii TableofContents
(cid:4)
§III.8 On (cid:2)(f(n)), f apolynomial ......................................... 85
n≤x
§III.9 Sumson(cid:2)(cid:8)(n),(cid:2)(cid:9)(n+k) ................................................. 85
§III.10 Inequalitiesconnecting(cid:2) ,d,(cid:7),(cid:4) .................................... 86
k
§III.11 Sumsover(cid:2)(p−1), paprime ......................................... 87
§III.12 On(cid:2)(mn) ................................................................... 87
§III.13 On(cid:2)(n)≥4(cid:1)(n) .......................................................... 88
§III.14 On(cid:2)(n+i)/(cid:2)(n+i −1)andrelatedtheorems .................... 88
§III.15 On(cid:2)((cid:2)(n));(cid:2)∗((cid:2)∗(n))and(cid:2)(k)(n),(cid:2)((cid:1)(n)),
(cid:1)((cid:2)(n)) ...................................................................... 89
§III.16 Divisibilitypropertiesof(cid:2) (n) .......................................... 91
k
§III.17 Divisibilityandcongruencespropertiesof(cid:2) (n) .................... 92
k
§III.18 Ons(n)=(cid:2)(n)−n ....................................................... 93
§III.19 Numberofdistinctvaluesof(cid:2)(n)/n,n ≤ x ......................... 94
§III.20 Frequencyofintegersm ≤ N withlog((cid:1)(m)/m)≤ x,
log((cid:2)(m)/m)≤ y .......................................................... 95
(cid:2)(an −1)
§III.21 On andrelatedfunctions ................................... 95
an −1
§III.22 Normalorderof(cid:6)((cid:2) (n)) ................................................ 96
k
§III.23 Numberofprimefactorsof((cid:2)(A ),A ) .............................. 97
k k
§III.24 On(cid:2)(pa)= xb ............................................................. 97
§III.25 Aninequalityfor(cid:2)∗(n) ................................................... 97
1
§III.26 Sumsover(cid:2)∗(n), ,(cid:2)∗2(n) ................................... 98
log(cid:2)∗(n) k
§III.27 Inequalitieson(cid:2)∗,d∗,(cid:2),(cid:3) ............................................. 99
k
§III.28 Thesumofexponentialdivisors ........................................ 99
§III.29 Averageorderof(cid:2)e(n) ................................................... 100
§III.30 Numberofdistinctprimedivisorsofanoddperfectnumber ..... 100
§III.31 Boundsfortheprimedivisorsofanoddperfectnumber .......... 102
§III.32 Densityofperfectnumbers .............................................. 104
§III.33 Multiplyperfectandmultiperfectnumbers ........................... 105
§III.34 k-perfectnumbers ......................................................... 106
§III.35 Primitiveabundantnumbers ............................................. 107
§III.36 Deficientnumbers ......................................................... 108
§III.37 Triperfectnumbers ........................................................ 108
§III.38 Quasiperfectnumbers..................................................... 109
§III.39 Almostperfectnumbers .................................................. 110
§III.40 Superperfectnumbers..................................................... 110
§III.41 Superabundantandhighlyabundantnumbers ........................ 111
§III.42 Amicablenumbers ........................................................ 112
§III.43 Weirdnumbers ............................................................. 113
TableofContents ix
§III.44 Hyperperfectnumbers .................................................... 114
§III.45 Unitaryperfectnumbers,bi-unitaryperfectnumbers ............... 114
§III.46 Primitiveunitaryabundantnumbers ................................... 115
§III.47 Nonunitaryperfectnumbers ............................................. 116
§III.48 Exponentiallyperfectnumbers.......................................... 116
§III.49 Exponentially,powerfulperfectnumbers ............................. 117
§III.50 Practicalnumbers .......................................................... 118
§III.51 Unitaryharmonicnumbers .............................................. 119
§III.52 PerfectGaussianintegers ................................................ 120
ChapterIV
P,p,B,β,ANDRELATEDFUNCTIONS.............................................. 121
§IV.1 Sumsover P(n), p(n), P(n)/p(n),1/Pr(n) .......................... 121
§IV.2 SumsoverlogP(n) ........................................................ 122
§IV.3 Sumsover P(n)−(cid:6)(n)and P(n)−(cid:4)(n) .................................... 123
§IV.4 Sumson1/p(n),(cid:6)(n)/p(n),d(n)/p(n) ............................... 123
§IV.5 Densityofreducibleintegers ............................................. 124
§IV.6 On p(n!+1), P(n!+1), P(F ) ........................................ 125
n
§IV.7 Greatestprimefactorofanarithmeticprogression ................... 125
§IV.8 P(n2+1)and P(n4+1) ................................................. 126
§IV.9 P(an −bn), P(ap−bp) .................................................. 127
§IV.10 P(u )forarecurrencesequence(u ) .................................. 128
n n
§IV.11 Greatestprimefactorofaproduct ...................................... 129
§IV.12 P(f(x)), f apolynomial ................................................. 130
§IV.13 Greatestprimefactorofaquadraticpolynomial ..................... 131
§IV.14 P(p+a), p(p+a), pprime ........................................... 132
§IV.15 On P(axm + byn) ........................................................ 132
§IV.16 Intervalscontainingnumberswithoutlargeprimefactors ......... 133
§IV.17 On P(n)/P(n+1) ........................................................ 134
§IV.18 Consecutiveprimedivisors .............................................. 135
§IV.19 Greatestprimefactorofconsecutiveintegers ........................ 135
§IV.20 Frequencyofnumberscontainingprimefactorsofacertain
relativemagnitude ......................................................... 136
§IV.21 Integerswithoutlargeprimefactors.Thefunction(cid:3)(x,y)
andDickman’sfunction .................................................. 136
§IV.22 Function(cid:3)(x,y;a,q).Integerswithoutlargeprimefactorsin
arithmeticprogressions ................................................... 141
§IV.23 On(n,(cid:9)(n))=1 ........................................................... 143
B(n)
§IV.24 Sumsover(cid:9) (n),B (n),B(n)−(cid:9)(n), ,
k k (cid:9)(n)
B(n) − (cid:9)(n)
............................................................... 143
P(n)
x TableofContents
(cid:9)(n) P(n)
§IV.25 Sumsover P(n), (cid:9)(n), B(n)− P1(n)−···− Pn−1(n) ........... 145
B(n)
§IV.26 Distributionof ....................................................... 146
(cid:9)(n)
§IV.27 On(−1)B(n) ................................................................. 146
§IV.28 SumsoverB1(n), P(n)/B1(n),B1(n)/B(n),1/B1(n),
etc. ............................................................................ 147
§IV.29 NumbersnwiththepropertyB(n)=B(n+1) ...................... 148
§IV.30 On(cid:1)greatestprimedivisorsofsumsofintegers ....................... 149
§IV.31 On f(P(n)), f acertainarithmeticfunction .................... 150
n≤x
§IV.32 On(cid:6)(x,y)andBuchstab’sfunction ................................... 151
§IV.33 Onthepartitionofprimesintotwosubsetswithnearlythe
samenumberofproducts ................................................. 153
ChapterV
(cid:6)(n),(cid:4)(n)ANDRELATEDFUNCTIONS............................................... 155
§V.1 Averageorderof(cid:6),(cid:4),(cid:4)−(cid:6),(cid:4) ....................................... 155
k
§V.2 Sumsover(cid:6)2(n),(cid:6)k(n) .................................................... 155
§V.3 Sumsover((cid:6)(n)−loglogx)2 ............................................ 156
(cid:1) 1 (cid:1) (cid:4)(n)
§V.4 , ,etc. .............................................. 157
(cid:6)(n) (cid:6)(n)
(cid:1)2≤n≤x 2≤n≤x
§V.5 (cid:6)k(p−1)(pprime) .................................................... 159
(cid:1)p≤n
§V.6 (cid:6)(f(p), f polynomial(pprime) .................................... 160
(cid:1)p≤n
§V.7 z(cid:6)(n)andrelatedsums .................................................. 161
n≤x
§V.8 Sumsover(cid:10)(n)=(−1)(cid:4)(n) ................................................ 162
§V.9 Sumsovern−1/(cid:6)(n),n−1/(cid:4)(n) ............................................... 162
§V.10 Sumsond(n)(cid:6)(n−1), d (n)(cid:6)(n) .................................... 163
k
(cid:6)(n) (cid:6)(n)
§V.11 Sumson , ....................................................... 163
P(n) (cid:9)(n)
§V.12 (cid:6)(a(n)),(cid:6)(d(n)),etc. ...................................................... 164
(cid:4)(n)−(cid:6)(n) (cid:4)(n)−(cid:6)(n)
§V.13 , ,etc. ........................................ 165
P(n) (cid:9)(n)
§V.14 Onthenumberofintegersn ≤ x with(cid:4)(n)−(cid:6)(n)=k ........... 165
§V.15 Estimatesoftype(cid:6)(n)≤c·logn/loglogn .......................... 167
§V.16 On(cid:6)(n)−(cid:6)(n+1)or(cid:6)(m)−(cid:6)(n) ................................... 168
§V.17 Thevaluesof(cid:6)onconsecutiveintegers ................................ 169
§V.18 Localgrowthof(cid:6)atconsecutiveintegers ............................. 170
§V.19 Normalorderof(cid:6)((cid:1)(n)) .................................................. 170
TableofContents xi
§V.20 Function(cid:6)(n;u,v) ......................................................... 171
§V.21 Onthenumberofvaluesn ≤ x with(cid:6)(n)> f(x) ................... 172
§V.22 On(cid:6)(2p−1),(cid:4)(an −1)/n .............................................. 172
§V.23 (cid:6)-highlycomposite,(cid:6)-largelycompositeand(cid:6)-interesting
numbers ....................................................................... 173
§V.24 On(cid:6)(n)/n .................................................................... 173
§V.25 On(n,(cid:6)(n))=1and(n,(cid:4)(n))=1 .................................... 174
§V.26 On(cid:6)((n,(cid:1)(n)))=k ........................................................ 174
§V.27 Gaussianlawoferrorsfor(cid:6) .............................................. 175
§V.28 Onthestatisticalpropertyofprimefactorsofnaturalnumbers
inarithmeticprogressions ................................................. 176
§V.29 Distributionofvaluesof(cid:6)inshortintervals .......................... 177
§V.30 Distributionof(cid:6)inthesieveofEratosthenes ......................... 177
§V.31 Numberofn ≤ x with(cid:4)(n)=i ......................................... 177
§V.32 Numberofn ≤ x with(cid:6)(n)=i ......................................... 180
§V.33 Thefunctions(cid:6)(n;E)andS(x,y;E,(cid:6)) ............................... 183
§V.34 Sumsetswithmanyprimefactors ........................................ 184
§V.35 Ontheintegersnforwhich(cid:4)(n)=k .................................. 185
ChapterVI
FUNCTIONµ;k-FREEANDk-FULLNUMBERS................................... 187
§VI.1 Averageorderof(cid:11)(n) ..................................................... 187
§VI.2 Estimatesfor M(x).Mertens’conjecture .............................. 187
§VI.3 (cid:11)inshortintervals ......................................................... 189
§VI.4 Sumsinvolving(cid:11)(n)with p(n)> y or P(n)< y,n ≤ x.
Squarefreenumberswithrestrictedprimefactors.................... 189
§VI.5 Oscillatorypropertiesof M(cid:1)(x)andrelatedresults ................... 190
§VI.6 Thefunction M(n,T)= (cid:11)(n) .................................. 192
d|n,d≤T
§VI.7 Mo¨biusfunctionoforderk ............................................... 193
§VI.8 Sumson(cid:11)(n)/n,(cid:11)(n)/n2,(cid:11)2(n)/n ................................... 194
§VI.9 Sumson(cid:11)(n)logn/n,(cid:11)(n)logn/n2 .................................. 195
§VI.10 Selberg’sformu(cid:5)la .(cid:6)......................................................... 196
x
§VI.11 Asumon(cid:11)(n) ....................................................... 197
n
§VI.12 Asumon(cid:11)(n)f(n)/n, f-multiplicative,0≤ f(p)≤1 .......... 197
§VI.13 Gandhi’sformula .......................................................... 197
§VI.14 Anextremalpropertyof(cid:11) ............................................... 198
§VI.15 OnasumconnectedwiththeMo¨biusfunction ...................... 199
(cid:11)2(n) (cid:11)2(n) (cid:11)2(n) (cid:11)(n)
§VI.16 Sumsover , , , ............................. 199
(cid:6)(n) (cid:6)2(n) (cid:1)(n) nd(n)
