Sandor j mitrinovic d s crstici b handbook of number theory vol 1 (ISBN 1402042159)( 2006)(637s) MT

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


A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10
ISBN-13
ISBN-10
ISBN-13

1-4020-4215-9 (HB)
978-1-4020-4215-7 (HB)
1-4020-3658-2 (e-book)
978-1-4020-3658-3 (e-book)

Published by Springer,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
www.springeronline.com

Printed on acid-free paper

1st ed. 1995. 2nd printing

All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Printed in the Netherlands.


TABLE OF CONTENTS
PREFACE .......................................................................................

xxv

BASIC SYMBOLS ............................................................................

1

BASIC NOTATIONS..........................................................................

2

Chapter I
EULER’S ϕ-FUNCTION.....................................................................
§ I. 1 Elementary inequalities for ␸ ..............................................
§ I. 2 Inequalities for ␸(mn) .......................................................
§ I. 3 Relations connecting ␸, ␴, d ...............................................
§ I. 4 Inequalities for Jk , ␴k , ␺ k ...................................................
§ I. 5 Unitary analogues of Jk , ␴k , d .............................................
§ I. 6 Composition of ␸, ␴, ␺ .....................................................
§ I. 7 Composition of ␴, ␸ .........................................................
§ I. 8 On the function n/␸(n) ......................................................
§ I. 9 Minimum of ␸(n)/n for consecutive values of n .......................
§ I.10 On ␸(n + 1)/␸(n) ...........................................................
§ I.11 On (␸(n + 1), ␸(n)) .........................................................
§ I.12 On (n, ␸(n)) ...................................................................
§ I.13 The difference of consecutive totients ...................................
§ I.14 Nonmonotonicity of ␸ . (A measure) ....................................
§ I.15 Nonmonotonicity of Jk .....................................................
§ I.16 Number of solutions of ␸(x) = n! ........................................
§ I.17 Number of solutions of ␸(x) = m ........................................
§ I.18 Number of values of ␸ less than or equal to x .........................
§ I.19 On composite n with ␸(n)|(n − 1) (Lehmer’s conjecture) ...........
§ I.20 Number of composite n ≤ x with ␸(n)|(n − 1) ........................
§ I.21
␸(n) ........................................................................

9
9
9
10

11
12
13
13
14
15
16
18
18
19
19
20
20
21
22
23
24
24

n≤x

§

I.22

f
k≤n

§

· ␸(k) ............................................................

25

3 2
x ........................................................
␲2

25

␸(n)/n ................................................................

27

␸(n) −

I.23 On
n≤x

§

k
n

I.24 On
n≤x


vi

Table of Contents

Jk (n) − x k+1 /(k + 1)␨ (k + 1) ...................................

28

An expansion of Jk ..........................................................
On n≤x 1/␸(n) and related questions ..................................
p≤x ␸( p − 1) for p prime ...............................................
On n≤x ␸( f (n)), f a polynomial .......................................
∗
n≤x ␸ (n),
n≤x ␸(n) ␸(n + k) and related results .................
Asymptotic formulae for generalized Euler functions ................
On ␸(x, n) = m≤x,(m,n)=1 1 and on Jacobstahl’s arithmetic
function ........................................................................
§ I.33 On the iteration of ␸ ........................................................
§ I.34 Iterates of ␸ and the order of ␸(k) (n)/␸(k+1) (n) ........................
§ I.35 Normal order of ␻(␸(n)) ...................................................

29
29
30
31
31
32
33
34
35
36

Chapter II
THE ARITHMETICAL FUNCTION d(n), ITS GENERALIZATIONS AND ITS
ANALOGUES..................................................................................
§ II. 1 The divisor functions at consecutive integers ..........................
§ II. 2 On d(n + i 1 ) > · · · > d(n + ir ) ..........................................
§ II. 3 Relations connecting d, ␻, , dk .........................................
§ II. 4 On d(mn) .....................................................................
§ II. 5 An inequality for dk (n) .....................................................
§ II. 6 Majorization for log d(n)/ log 2 ..........................................
§ II. 7 max d(n) and max(d(n), d(n + 1)) and generalizations ..............

39
39
40
41
42
42
42
44

§

I.25 On
n≤x

§

I.26
I.27

§ I.28
§ I.29
§ I.30
§ I.31
§ I.32
§

n≤x

§

n≤x

II. 8 Highly composite, superior highly composite, and largely
composite numbers ..........................................................
§ II. 9 Congruence property of d(n) ..............................................
§ II.10
(x) =
d(n) − x log x − (2␥ − 1)x ...............................

45
47
47

n≤x

§

d( p − 1), p prime .....................................................

II.11

49

p≤x

dk (n) − x · Pk−1 (log x), k ≥ 2 ...........................

51

dk2 (n) .....................................................................

55

(g ∗ dk ) (n) ..........................................................

55

II.15 3 (x) ..........................................................................
II.16 The divisor problem in arithmetic progressions ......................
§ II.17 On
1/dk (n) ..............................................................

56
57
59

§

II.12

k (x)

=
n≤x

§

II.13
n≤x

§
§

II.14 On
n≤x

§

n≤x


Table of Contents

vii

§

II.18 Average order of dk (n) over integers free of large prime
factors .........................................................................

§ II.19 On a sum on dk and Legendre’s symbol ................................
§ II.20 A sum on dk , d and ␻ ......................................................
§ II.21 On
d(n) · d(n + N ) and related problems .........................

60
60
61
61

n≤x

dk (n) · d(n + 1) and related questions .........................

63

II.23 Iteration of d .................................................................
II.24 On d( f (n)) and d(d( f (n))), f a polynomial ..........................
§ II.25 On
d(n 2 + a) and
d(m 2 + n 2 ) ................................

65
66
67

§

II.22 On
n≤x

§
§

n≤x

§
§
§

II.26

m,n≤x

d(| f (r, s)|), f (x, y) a binary cubic form ....................

68

II.27 Weighted divisor problem .................................................
II.28 On
d(n − k t ) ..........................................................

68
69

| f (r,s)|≤N

§

II.29

§ II.30
§ II.31
§ II.32
§ II.33

k
Divisor sums on squarefree or squarefull integers ....................
Exponential divisors .......................................................
Bi-unitary divisors ..........................................................
Sums over d(n) · ␻(n), d(n)/␻(n), ␻(d(n)), (d(n)) ................
d(a(n)), a(n) the number of abelian groups with n

69
71
72
72

n≤x

elements ......................................................................
d(n) in short intervals ......................................................
Number of distinct values of d(n) for 1 ≤ n ≤ x .....................
On the distribution function of d(n) .....................................
On (n d(n), ␴(n)) = 1 ......................................................
Average value for the number of divisors of sums a + b ............

73
73
74

74
75
75

Chapter III
SUM-OF-DIVISORS FUNCTION, GENERALIZATIONS, ANALOGUES;
PERFECT NUMBERS AND RELATED PROBLEMS ................................
§ III. 1 Elementary inequalities on ␴(n) and ␴(n)/n .........................
§ III. 2 On ␴(n)/n log log n ........................................................
§ III. 3 On ␴k (n)/n k .................................................................
§ III. 4
␴(n),
␴(n),
␴(n) ...................................

77
77
79
80
81

§

II.34
§ II.35
§ II.36
§ II.37
§ II.38

n≤x

n≤x, p|n

n≤x,(n,k)=1

␴(n)
§ III. 5 Sums over
............................................................
n
§ III. 6 Sums over ␴ k (n) ............................................................
§ III. 7 On sums over ␴−␣ ( f (n)), f a polynomial (0 < ␣ < 1) .............

82
83
84


viii

Table of Contents

␴( f (n)), f a polynomial .........................................

85

III. 9 Sums on ␴␣ (n), ␴␤ (n + k) .................................................
III.10 Inequalities connecting ␴k , d, ␥ , ....................................
§ III.11 Sums over ␴( p − 1), p a prime .........................................
§ III.12 On ␴(mn) ...................................................................
§ III.13 On ␴(n) ≥ 4␸(n) ..........................................................

§ III.14 On ␴(n + i)/␴(n + i − 1) and related theorems ....................
§ III.15 On ␴(␴(n)); ␴ ∗ (␴ ∗ (n)) and ␴ (k) (n), ␴(␸(n)),
␸(␴(n)) ......................................................................
§ III.16 Divisibility properties of ␴k (n) ..........................................
§ III.17 Divisibility and congruences properties of ␴k (n) ....................
§ III.18 On s(n) = ␴(n) − n .......................................................
§ III.19 Number of distinct values of ␴(n)/n, n ≤ x .........................
§ III.20 Frequency of integers m ≤ N with log(␸(m)/m) ≤ x,
log(␴(m)/m) ≤ y ..........................................................
␴(a n − 1)
§ III.21 On
and related functions ...................................
an − 1
§ III.22 Normal order of ␻(␴k (n)) ................................................
§ III.23 Number of prime factors of (␴(Ak ), Ak ) ..............................
§ III.24 On ␴( p a ) = x b .............................................................
§ III.25 An inequality for ␴ ∗ (n) ...................................................
1
2
§ III.26 Sums over ␴ ∗ (n),
, ␴k∗ (n) ...................................
∗
log ␴ (n)
§ III.27 Inequalities on ␴k∗ , d ∗ , ␴, ␺ .............................................
§ III.28 The sum of exponential divisors ........................................
§ III.29 Average order of ␴ e (n) ...................................................
§ III.30 Number of distinct prime divisors of an odd perfect number .....
§ III.31 Bounds for the prime divisors of an odd perfect number ..........
§ III.32 Density of perfect numbers ..............................................
§ III.33 Multiply perfect and multiperfect numbers ...........................

§ III.34 k-perfect numbers .........................................................
§ III.35 Primitive abundant numbers .............................................
§ III.36 Deficient numbers .........................................................
§ III.37 Triperfect numbers ........................................................
§ III.38 Quasiperfect numbers .....................................................
§ III.39 Almost perfect numbers ..................................................
§ III.40 Superperfect numbers .....................................................
§ III.41 Superabundant and highly abundant numbers ........................
§ III.42 Amicable numbers ........................................................
§ III.43 Weird numbers .............................................................

85
86
87
87
88
88

§
§
§

III. 8 On
n≤x

89
91
92
93
94

95
95
96
97
97
97
98
99
99
100
100
102
104
105
106
107
108
108
109
110
110
111
112
113


Table of Contents
§

III.44

III.45
§ III.46
§ III.47
§ III.48
§ III.49
§ III.50
§ III.51
§ III.52
§

ix

Hyperperfect numbers ....................................................
Unitary perfect numbers, bi-unitary perfect numbers ...............
Primitive unitary abundant numbers ...................................
Nonunitary perfect numbers .............................................
Exponentially perfect numbers ..........................................
Exponentially, powerful perfect numbers .............................
Practical numbers ..........................................................
Unitary harmonic numbers ..............................................
Perfect Gaussian integers ................................................

Chapter IV
P, p, B, β, AND RELATED FUNCTIONS ..............................................
§ IV. 1 Sums over P(n), p(n), P(n)/ p(n), 1/P r (n) ..........................
§ IV. 2 Sums over log P(n) ........................................................
§ IV. 3 Sums over P(n)−␻(n) and P(n)− (n) ....................................
§ IV. 4 Sums on 1/ p(n), ␻ (n)/ p(n), d(n)/ p(n) ...............................
§ IV. 5 Density of reducible integers .............................................
§ IV. 6 On p(n! + 1), P(n! + 1), P(Fn ) ........................................

§ IV. 7 Greatest prime factor of an arithmetic progression ...................
§ IV. 8 P(n 2 + 1) and P(n 4 + 1) .................................................
§ IV. 9 P(a n − bn ), P(a p − b p ) ..................................................
§ IV.10 P(u n ) for a recurrence sequence (u n ) ..................................
§ IV.11 Greatest prime factor of a product ......................................
§ IV.12 P( f (x)), f a polynomial .................................................
§ IV.13 Greatest prime factor of a quadratic polynomial .....................
§ IV.14 P( p + a), p( p + a), p prime ...........................................
§ IV.15 On P(ax m + by n ) ........................................................
§ IV.16 Intervals containing numbers without large prime factors .........
§ IV.17 On P(n)/P(n + 1) ........................................................
§ IV.18 Consecutive prime divisors ..............................................
§ IV.19 Greatest prime factor of consecutive integers ........................
§ IV.20 Frequency of numbers containing prime factors of a certain
relative magnitude .........................................................
§ IV.21 Integers without large prime factors. The function ␺ (x, y)
and Dickman’s function ..................................................
§ IV.22 Function ␺ (x, y; a, q). Integers without large prime factors in
arithmetic progressions ...................................................
§ IV.23 On (n, ␤(n)) = 1 ...........................................................
B(n)
§ IV.24 Sums over ␤k (n), Bk (n), B(n) − ␤(n),
,
␤(n)
B(n) − ␤(n)
...............................................................
P(n)

114
114

115
116
116
117
118
119
120

121
121
122
123
123
124
125
125
126
127
128
129
130
131
132
132
133
134
135
135
136
136

141
143

143


x

Table of Contents

␤(n) P(n)
,
, B(n) − P1 (n) − · · · − Pn−1 (n) ...........
P(n) ␤(n)
B(n)
Distribution of
.......................................................
␤(n)
On (−1)B(n) .................................................................
Sums over B1 (n), P(n)/B1 (n), B1 (n)/B(n), 1/B1 (n),
etc. ............................................................................
Numbers n with the property B(n) = B(n + 1) ......................
On greatest prime divisors of sums of integers .......................
On
f (P(n)), f a certain arithmetic function ....................

§

IV.25 Sums over

145

§

IV.26

146

§

IV.27
IV.28

§
§

IV.29
§ IV.30
§ IV.31

146
147
148
149
150

n≤x

§

IV.32 On (x, y) and Buchstab’s function ................................... 151
§ IV.33 On the partition of primes into two subsets with nearly the
same number of products ................................................. 153

Chapter V
␻(n), (n) AND RELATED FUNCTIONS...............................................
§ V. 1 Average order of ␻, ,
− ␻, k .......................................
§ V. 2 Sums over ␻2 (n), ␻k (n) ....................................................
§ V. 3 Sums over (␻(n) − log log x)2 ............................................
1
(n)
§ V. 4
,
, etc. ..............................................
␻(n)
␻(n)
2≤n≤x
2≤n≤x
§

V. 5

155
155
155
156
157

␻k ( p − 1) ( p prime) ....................................................

159

␻( f ( p), f polynomial ( p prime) ....................................

160

p≤n

§

V. 6
p≤n

§

z ␻(n) and related sums .................................................. 161

V. 7
n≤x

V. 8 Sums over ␭(n) = (−1) (n) ................................................
§ V. 9 Sums over n −1/␻(n) , n −1/ (n) ...............................................
§ V.10 Sums on d(n) ␻(n − 1), dk (n) ␻(n) ....................................
␻(n) ␻(n)
§ V.11 Sums on
,
.......................................................
P(n) ␤(n)
§ V.12 ␻(a(n)), ␻(d(n)), etc. ......................................................

(n) − ␻(n)
(n) − ␻(n)
§ V.13
,
, etc. ........................................
P(n)
␤(n)
§ V.14 On the number of integers n ≤ x with (n) − ␻(n) = k ...........
§ V.15 Estimates of type ␻(n) ≤ c · log n/ log log n ..........................
§ V.16 On ␻(n) − ␻(n + 1) or ␻(m) − ␻(n) ...................................
§ V.17 The values of ␻ on consecutive integers ................................
§ V.18 Local growth of ␻ at consecutive integers .............................
§ V.19 Normal order of ␻(␸(n)) ..................................................
§

162
162
163
163
164
165
165
167
168
169
170
170


Table of Contents

§

V.20
V.21
§ V.22
§ V.23
§

§

V.24
§ V.25
§ V.26
§ V.27
§ V.28
§

V.29
§ V.30
§ V.31
§ V.32
§ V.33
§ V.34
§ V.35

xi

Function ␻(n; u, v) .........................................................
On the number of values n ≤ x with ␻(n) > f (x) ...................
On ␻(2 p − 1), (a n − 1)/n ..............................................

␻-highly composite, ␻-largely composite and ␻-interesting
numbers .......................................................................
On ␻(n)/n ....................................................................
On (n, ␻(n)) = 1 and (n, (n)) = 1 ....................................
On ␻((n, ␸(n))) = k ........................................................
Gaussian law of errors for ␻ ..............................................
On the statistical property of prime factors of natural numbers
in arithmetic progressions .................................................
Distribution of values of ␻ in short intervals ..........................
Distribution of ␻ in the sieve of Eratosthenes .........................
Number of n ≤ x with (n) = i .........................................
Number of n ≤ x with ␻(n) = i .........................................
The functions ␻(n; E) and S(x, y; E, ␻) ...............................
Sumsets with many prime factors ........................................
On the integers n for which (n) = k ..................................

Chapter VI
FUNCTION µ; k-FREE AND k-FULL NUMBERS ...................................
§ VI. 1 Average order of ␮(n) .....................................................
§ VI. 2 Estimates for M(x). Mertens’ conjecture ..............................
§ VI. 3 ␮ in short intervals .........................................................
§ VI. 4 Sums involving ␮(n) with p(n) > y or P(n) < y, n ≤ x.
Squarefree numbers with restricted prime factors ....................
§ VI. 5 Oscillatory properties of M(x) and related results ...................
§ VI. 6 The function M(n, T ) =
␮(n) ..................................

171
172
172

173
173
174
174
175
176
177
177
177
180
183
184
185

187
187
187
189
189
190
192

d|n,d≤T

§

VI. 7 M¨obius function of order k ...............................................
§ VI. 8 Sums on ␮(n)/n, ␮(n)/n 2 , ␮2 (n)/n ...................................
§ VI. 9 Sums on ␮(n) log n/n, ␮(n) log n/n 2 ..................................
§ VI.10 Selberg’s formula ..........................................................

x
§ VI.11 A sum on ␮(n)
.......................................................
n
§ VI.12 A sum on ␮(n) f (n)/n, f -multiplicative, 0 ≤ f ( p) ≤ 1 ..........
§ VI.13 Gandhi’s formula ..........................................................
§ VI.14 An extremal property of ␮ ...............................................
§ VI.15 On a sum connected with the M¨
obius function ......................
␮2 (n) ␮2 (n) ␮2 (n) ␮(n)
§ VI.16 Sums over
,
,
,
.............................
␻(n) ␻2 (n) ␸(n) nd(n)

193
194
195
196
197
197
197
198
199
199


xii

Table of Contents
§

VI.17 The distribution of integers having a given number of prime
factors .......................................................................
§ VI.18 Number of squarefree integers ≤ x ....................................
§ VI.19 On squarefree integers ....................................................
§ VI.20 Intervals containing a squarefree integer ..............................
§ VI.21 Distribution of squarefree numbers ....................................
§ VI.22 On the frequency of pairs of squarefree numbers ...................
§ VI.23 Smallest squarefree integer in an arithmetic progression ..........
§ VI.24 The greatest squarefree divisor of n ....................................
§ VI.25 Estimates involving the greatest squarefree divisor of n ...........
§ VI.26 Estimates for N (x, y) = card {n ≤ x : ␥ (n) ≤ y} ..................
§ VI.27 Number of non-squarefree odd, positive integers ≤ x ..............
§ VI.28 Number of squarefree numbers ≤ X which are quadratic
residues (mod p) ...........................................................
§ VI.29 Squarefree integers in nonlinear sequences ...........................
§ VI.30 Sumsets containing squarefree and k-free integers ..................
§ VI.31 On the M¨
obius function ..................................................
§ VI.32 Number of k-free integers ≤ x ..........................................
§ VI.33 Number of k-free integers ≤ x, which are relatively prime to
n ..............................................................................
§ VI.34 Schnirelmann density of the k-free integers ..........................
§ VI.35 Powerfree integers represented by linear forms .....................
§ VI.36 On the power-free value of a polynomial .............................
§ VI.37 Number of r -free integers ≤ x that are in arithmetic
progression .................................................................

§ VI.38 Squarefree numbers as sums of two squares .........................
§ VI.39 Distribution of unitary k-free integers .................................
§ VI.40 Counting function of the (k, r )-integers ...............................
§ VI.41 Asymptotic formulae for powerful numbers .........................
§ VI.42 Maximal k-full divisor of an integer ...................................
§ VI.43 Number of squarefull integers between successive squares .......
Chapter VII
FUNCTION π(x), ψ(x), θ(x), AND THE SEQUENCE OF PRIME NUMBERS
§ VII. 1 Estimates on ␲(x). Chebyshev’s theorem. The prime
number theorem ...........................................................
x
dy
§ VII. 2 Approximation of ␲(x) by
..................................
log
y
2
§ VII. 3 On ␲(x) − li x. Sign changes ...........................................
§ VII. 4 On ␲(x) − ␲(x − y) for y = x ␪ .......................................
§ VII. 5 On ␲(x + y) ≤ ␲(x) + ␲(y) ............................................
§ VII. 6 On
(␲ ∗ (k) − ␲(k)) .................................................
q≤k≤n

200
201
202
202
204
205

206
208
209
210
210
211
211
212
213
213
216
217
218
218
220
221
221
222
222
226
226

227
227
228
229
232
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237

Table of Contents

xiii

1
............................................................
␲(n)
§ VII. 8 Number of primes p ≤ x for which p + k is a prime and
related questions ...........................................................
§ VII. 9 Number of primes p ≤ x with ␻( p + 2) ≤ 2 ........................
§ VII.10 Almost primes P2 in intervals ..........................................
§ VII.11 P21 in short intervals .....................................................
§ VII.12 Consecutive almost primes .............................................
§ VII.13 Primes in short intervals .................................................
§ VII.14 Primes between x and a · x, (a > 1, constant). Bertrand’s
postulate ....................................................................
§ VII.15 On intervals containing no primes ....................................
§ VII.16 Difference between consecutive primes ..............................
§ VII.17 Comparison of p1 . . . pn with pn+1 ...................................
§ VII.18 Elementary estimates on p[an] , pmn , pn+1 / pn .......................
§ VII.19 Sharp upper and lower bounds for pn ................................
§ VII.20 The nth composite number .............................................
√
√
§ VII.21 On infinite series involving
pn+1 − pn , 1/n( pn+1 − pn )
and related problems .....................................................
§ VII.22 Largest gap between consecutive primes below x ..................
§ VII.23 On min(dn , dn+1 ) and various sums over dn .........................

§ VII.24 On the sign changes of dn − dn+1 and related theorems on
primes ......................................................................
§ VII.25 The sequence (bn ) defined by bn = dn / log pn ......................
§ VII.26 Results on pk /k ..........................................................
§ VII.27 On the sums of prime powers ..........................................
1
§ VII.28 Estimates on
......................................................
p≤x p
1
§ VII.29 Estimates on
1−
.............................................
p
p≤x
§ VII.30 Some properties of ␺ -function ........................................
§ VII.31 Selberg’s formula .........................................................
§ VII.32 On
␺ (n) ...............................................................
§

VII. 7 A sum on

VII.33
§ VII.34
§ VII.35
§ VII.36
§ VII.37

238

240
240
241
242
243
243
245
245
246
247
247
247
248
249
250
253
254
256
257
257
259
259
262
263

n≤x

Estimates on ␺ (x + h) − ␺ (x) ........................................
On (x) = ␺ (x) − x ....................................................
Results on ␪(x) ...........................................................

Primes in short intervals .................................................
Estimates concerning (n) and certain generalizations.
Sign-changes in the remainder .........................................
§ VII.38 A sum over 1/ (n) ......................................................
§ VII.39 On Chebyshev’s conjecture .............................................
§

238

263
264
267
270
270
273
273


xiv

Table of Contents
§

VII.40 A sum involving primes ................................................. 274

Chapter VIII
PRIMES IN ARITHMETIC PROGRESSIONS AND OTHER SEQUENCES ....
§ VIII. 1 Dirichlet’s theorem on arithmetic progressions ....................
§ VIII. 2 Bertrand’s and related problems in arithmetic progressions .....
§ VIII. 3 Sums over 1/ p, log p/ p when p ≤ x, p ≡ l(mod k) ............

§ VIII. 4 The n-th prime in an arithmetic progression ........................
§ VIII. 5 Least prime in an arithmetic progression. Linnik’s theorem.
Various estimates on p(k, l) ............................................
§ VIII. 6 Siegel-Walfisz theorem. The Bombieri-Vinogradov theorem ....
§ VIII. 7 Primes in arithmetic progressions .....................................
§ VIII. 8 Bombieri’s theorem in short intervals ................................
§ VIII. 9 Prime number theorem for arithmetic progressions ...............
§ VIII.10 An estimate on ␲(x; p, −1) ...........................................
§ VIII.11 Assertions equivalent to the prime number theorem for
li x
arithmetic progressions. Sums over ␲(x; k, l) −
...........
␸(k)
§ VIII.12 Brun-Titchmarsh theorem .............................................
§ VIII.13 Application of the Brun-Titchmarsh theorem on lower
bounds for ␲(x; k, l) ....................................................
§ VIII.14 On ␲(x + x ␪ ; k · l) − ␲(x; k, l) .......................................
§ VIII.15 Barban’s theorem ........................................................
§ VIII.16 On generalizations of the Bombieri-Vinogradov theorem .......
§ VIII.17 An upper bound for ␲k (y; k, l) = number of primes
x < p ≤ x + y with p ≡ l(mod k) ..................................
§ VIII.18 An analogue of the Brun-Titchmarsh inequality ...................
§ VIII.19 On Goldbach-Vinogradov’s theorem. The prime k-tuple
conjecture on average ...................................................
x 2
li x 2
§ VIII.20 Sums over ␪(x; k, l) −
, ␲(x; k, l) −
...........
␸(k)

␸(k)
§ VIII.21 Oscillation theorems for primes in arithmetic progressions .....
§ VIII.22 Special results on finite sums over primes ..........................
§ VIII.23 Infinitely many sets of three distinct primes and an almost
prime in arithmetic progressions .....................................
§ VIII.24 Large prime factors of integers in an arithmetic progression ....
§ VIII.25 Almost primes in arithmetic progressions ..........................
§ VIII.26 Arithmetic progressions that consist only in primes ..............
§ VIII.27 Number of n ≤ x such that there is no prime between n 2
and (n + 1)2 ..............................................................
§ VIII.28 Primes in the sequence [n c ] ...........................................
§ VIII.29 Number of primes p ≤ x for which [ p c ] is prime ................
§ VIII.30 Almost primes in (n 2 + 1) and related sequences .................

275
275
275
276
278
278
280
283
283
285
285
286
287
290
290
291

291
292
292
293
294
295
297
297
298
299
299
299
300
301
302


Table of Contents

VIII.31 Primes p ≤ N of the form p = [c n] ................................
VIII.32 Primes of the form n · 2n + 1 or p · 2 p + 1 or 2 p ± p ...........
§ VIII.33 Primes of the form x 2 + y 2 + 1 ......................................
log p
§ VIII.34 On a sum on
when p ∈ L = arithmetic progression ......
p
§ VIII.35 Recurrent sequences of primes .......................................
§ VIII.36 Composite values of exponential and related sequences .........
§ VIII.37 Primes in partial sums of n n ...........................................
§ VIII.38 Beurling’s generalized integers .......................................

§ VIII.39 Accumulation theorems for primes in arithmetic progressions .......................................................................
§ VIII.40 About the Shanks-R´enyi race problem ..............................
§
§

Chapter IX
ADDITIVE AND DIOPHANTINE PROBLEMS INVOLVING PRIMES .........
§ IX. 1 Schnirelman’s theorem. Vinogradov’s theorem ......................
§ IX. 2 Number of representations of N in the form p1n + · · · + pkn .
Vinogradov’s three primes theorem .....................................
§ IX. 3 R´enyi’s theorem. Chen’s theorem .......................................
§ IX. 4 Improvements on Chen’s theorem ......................................
§ IX. 5 On number of writings of N as ␧1 . . . ␧s + p1 . . . pr or
␧1 . . . ␧s + p1 . . . pr +1 . A common generalization of Chen’s and
Linnik’s theorems ..........................................................
§ IX. 6 On p1k + p2k = N . Estimates on the number of solutions.
Binary Hardy-Littlewood problem ......................................
§ IX. 7 Number of Goldbach numbers and related problems ................
§ IX. 8 The exceptional set in Goldbach’s problem ...........................
§ IX. 9 Partitions into primes ......................................................
§ IX.10 Partitions of n into parts, or distinct parts in a set A ................
§ IX.11 Representations in the form k = ap1 + · · · + ar pr ( pi primes)
with restricted primes pi .................................................
§ IX.12 Representations in the form N = p + n, p prime, with certain
restrictions on n ............................................................
§ IX.13 On integers of the form p + a k ( p prime, a > 1) or p 2 + a k
or p + q! (q prime), etc. .................................................
§ IX.14 Linnik’s theorem (on the Hardy-Littlewood problem) .............
§ IX.15 Representations in the form p13 + p23 + p33 + x 3 ( pi primes),
etc. ............................................................................

§ IX.16 Number of solutions of n = p + x y ( p prime; x, y ≥ 1) ..........
§ IX.17 Representations of primes by quadratic forms .......................
§ IX.18 Number of solutions of m = p1 + v a , n = p2 + v a , (m < x,
n < x, pi primes) ..........................................................

xv

304
305
306
306
307
307
308
308
309
311

313
313
314
316
317

318
320
321
323
324
326

327
327
328
330
332
332
333
333


xvi

Table of Contents
§

IX.19 Number of representations of n as the sum of the square of a
prime and an r -free integer ..............................................
§ IX.20 Distinct integers ≤ x which can be expressed as p + a ki ,
where (ki ) is a certain sequence ........................................
§ IX.21 Waring-Goldbach-type problems for the function f (x) = x c ,
c > 12. Hybrid of theorems by Vinogradov and
ˇ
Pjatecki˘ı-Sapiro
............................................................
§ IX.22 Integers not representable in the form p + [n c ] (c > 1) ...........
§ IX.23 On the maximal distance between integers composed of small
primes .......................................................................
§ IX.24 On the representation of N as N = a + b or N = a + b + c
with restrictions on P(ab) or P(abc) ..................................
§ IX.25 On the maximal length of two sequences of consecutive

integers with the same prime divisors .................................
p+1
§ IX.26 Representation of n as n =
( p, q primes) ....................
q +1
§ IX.27 An additive property of squares and primes ..........................
1
√
§ IX.28 On the distribution of { p} and { p ␪ },
≤ ␪ ≤ 1 ..................
2
§ IX.29 Diophantine approximations by almost primes ......................
§ IX.30 Number of solutions of f ( p) < p −␶ +␧ ( p prime) ...............
§ IX.31 A sum involving p ␣ ( p prime) ......................................
§ IX.32 On the distribution of ␣ p modulo one ................................
§ IX.33 Simultaneous diophantine approximation with primes .............
§ IX.34 Diophantine approximation by prime numbers ......................
§ IX.35 Metric diophantine approximation with two restricted prime
variables .....................................................................
§ IX.36 The uniform distributed sequences (␣ p) and ( p ␣ ), where
0 < ␣ < 1, and ( p ␥ ), ␥ > 1, ␥ = integer ............................

334
334

335
336
336
337
339

339
341
342
343
343
344
344
345
346
347
348

Chapter X
EXPONENTIAL SUMS...................................................................... 349
§ X. 1 Basic estimates on
e(m ␣) ............................................ 349
§

n≤x

X. 2
§ X. 3
§ X. 4
§ X. 5
§ X. 6

Weyl’s method ...............................................................
Van der Corput’s method ..................................................
Vinogradov’s method .......................................................
Theory of exponent pairs ..................................................

Multiple trigonometric sums ..............................................
b

§

X. 7 Estimates on

g(t) · ei f (t) dt ............................................

349
350
353
353
355
356

c

§

ei f (x,y) dx dy or

X. 8 Estimates of type
D

e( f (n, m))
(n, m)∈D

where D is a plane domain ................................................

357


Table of Contents

xvii

§

X. 9 Vinogradov’s mean-value theorem ....................................... 359
X.10 Exponential sums containing primes ................................... 360
§ X.11 Exponential sums of type
(m + w)ti .......................... 361
§

M ≤m≤M

§

X.12 Complete trigonometric sums ............................................
§ X.13 Nearly complete and supercomplete rational trigonometric
sums ...........................................................................
§ X.14 Hua’s estimate ..............................................................
§ X.15 Gaussian sums ..............................................................
§ X.16 Estimates by Linnik and Vinogradov ...................................
§ X.17 Sums of type
(log p) · e(ap k /q) ( p prime) and

362
364

365
366
366

p≤N

a
≤ q12 for (a, q) = 1 ........................
q
p≤N
Estimates of trigonometric sums over primes in short intervals ...
A short exponential rational trigonometric sum ......................
Estimates on sums over e(uh/k), when f (u) ≡ 0(mod k),
0 < u ≤ k and k ≤ x ......................................................
Exponential sums formed with the M¨obius function ................
On
␮2 (n)e(␣n 3 ) ........................................................
e( p␣) where ␣ −

§

X.18
X.19
§ X.20
§

§

X.21
§ X.22

§

369
371
372
372
373

n≤x

X.23 The sum of e(␣n), when (n) = k ......................................
§ X.24 Exponential sums involving the Ramanujan function ...............
§ X.25 An exponential sum involving r (n) (number of
representations of n as a sum of two squares) .........................
§ X.26 Exponential sums on integers having small prime factors ..........
√
§ X.27 A result on
e(x n) ...................................................
§

367

374
374
375
375
376

n≤N

X.28 Kloosterman sums. Sali´e’s and Weil’s estimates .....................
§ X.29 Exponential sums connected with the distribution of
␣p(mod 1) and with diophantine approximation with primes
or almost primes ............................................................
§ X.30 On e(␣x 3 ) ....................................................................
§ X.31 Exponential sums and the logarithmic uniform distribution of
(␣n + ␤ log n) ...............................................................
§ X.32 Exponential sums with multiplicative coefficients ...................
§ X.33 On
(u) (␯)e( f (u␯)) ................................................
§ X.34 Exponential sums involving quadratic polynomials and
sequences ....................................................................
§ X.35 The large sieve as an estimate for exponential sums .................
§ X.36 An estimate for the derivative of a trigonometric polynomial ......
§ X.37 Weighted exponential sums and discrepancy ..........................
§ X.38 Deligne’s estimates .........................................................
§ X.39 On fourth moments of exponential sums ...............................

377

378
379
380
381
382
383
383
386
386

386
387


xviii

Table of Contents
§

X.40 Biquadratic Weyl sums ....................................................

Chapter XI
CHARACTER SUMS.........................................................................
§ XI. 1 P´
olya-Vinogradov inequality and a generalization. Character
sums modulo a prime power. Burgess’ estimate ......................
§ XI. 2 On the constant in the P´
olya-Vinogradov inequality. Large
values of character sums ..................................................
§ XI. 3 Burgess’ character sum estimate ........................................
§ XI. 4 A character sum estimate for nonprincipal character
␹ (mod q) ....................................................................
§ XI. 5 A sum on ␹ (u + v), on sets with no two integers of which
are congruent ................................................................
§ XI. 6 A lower bound on a character sum estimate arising in a
problem concerning the distribution of sequences of integers in
arithmetic progressions ...................................................
§ XI. 7 Powers of character sums .................................................
§ XI. 8 Sums of characters with primes. Vinogradov’s theorem ............
§ XI. 9 Distribution of pairs of residues and nonresidues of special

form ...........................................................................
§ XI.10 A character sum estimate involving ␮(n) and (n) .................
§ XI.11 An upper bound for a character sum involving (n) ...............
§ XI.12 Half Gauss sums ...........................................................
§ XI.13 Exponential sums with characters. A large-sieve density
estimate ......................................................................
q−1

§

␹ (n) · k n ...........................................................

XI.14 On

387

389
389
390
393
393
394

394
394
396
397
397
398
398

399
400

k=1
M

␹ (x) · e(ax/ p) ....................................

401

XI.16 An infinite series of characters with application to zero
density estimates for ᏸ functions ......................................
§ XI.17 Character sums of polynomials .........................................
§ XI.18 Quadratic character of a polynomial ...................................
§ XI.19 Distribution of values of characters in sparse sequences ...........
§ XI.20 Estimation of character sums modulo a power of a prime .........
§ XI.21 Mean values of character sums .........................................
§ XI.22 On
␹ (n), with S(x, y) = {n ≤ x : P(n) ≤ y} ...............

402
402
403
404
404
406
407

§

XI.15 Estimates on
x=N +1

§

n∈S(x,y)

§

XI.23
§ XI.24
§ XI.25
§ XI.26

Large sieve-type inequalities via character sum estimates .........
Large sieve-type inequalities of Selberg and Motohashi ...........
A large sieve density estimate ...........................................
A theorem by Wolke ......................................................

407
409
410
410


Table of Contents
§

XI.27 Character sums involving

xix

(X, ␹ ) =

(n) ␹ (n) ............... 411
n≤x

XI.28 An estimate involving ␹ 1 ∗ ␹ 2 ...........................................
XI.29 Number of primitive characters mod n, and the number of
characters with modulus ≤ x ............................................
§ XI.30 Continuous additive characters of a topological abelian group ...
§ XI.31 An estimate for perturbed Dirichlet characters ......................
§ XI.32 Estimates on Hecke characters ..........................................
§ XI.33 Character sums in finite fields ...........................................
§ XI.34 Gauss sums, Kloosterman sums ........................................
§ XI.35 Dirichlet characters on additive sequences ...........................
§

411

§

Chapter XII
BINOMIAL COEFFICIENTS, CONSECUTIVE INTEGERS AND RELATED
PROBLEMS ....................................................................................
n
§ XII. 1 On p a
................................................................
k
§ XII. 2 Number of binomial coefficients not divisible by an integer ......

§ XII. 3 Number of distinct prime factors of binomial coefficients ........
2n
§ XII. 4 Divisibility properties of
........................................
n
2n
§ XII. 5 Squarefree divisors of
...........................................
n
§ XII. 6 Divisibility properties of consecutive integers .......................
§ XII. 7 The theorem of Sylvester and Schur ...................................
§ XII. 8 On the prime factorization of binomial coefficients ................
§ XII. 9 Inequalities and estimates involving binomial coefficients ........
§ XII.10 On unimodal sequences of binomial coefficients ...................
§ XII.11 A theorem of Pillai and Szekeres ......................................
§ XII.12 A sum on a function connected with consecutive integers ........
§ XII.13 On consecutive integers. Theorems of Erd˝
os-Rankin and
Shorey ......................................................................
§ XII.14 On prime factors on consecutive integers ............................
§ XII.15 The Grimm conjecture and analogues problems ...................
§ XII.16 Great values of a function connected with consecutive
integers .....................................................................
§ XII.17 A theorem of Erd˝
os and Selfridge on the product of consecutive integers ................................................................
§ XII.18 Products terms in an arithmetical progression ......................
§ XII.19 On the sequence n! + k, 2 ≤ k ≤ n ...................................
§ XII.20 Decomposition of n! into prime factors ..............................
§ XII.21 Divisibility of products of factorials ..................................
§ XII.22 Powers and factorials ....................................................

§ XII.23 Distribution of divisors of n! ...........................................

412
413
413
413
414
415
416

417
417
418
419
422
424
425
426
427
430
434
435
436
436
437
438
440
440
441
442

442
444
445
447


xx

Table of Contents
§

XII.24 Stirling’s formula and power of factorials ...........................
XII.25 The Wallis sequence and related inequalities on gamma
function ....................................................................
§ XII.26 A special sequence of Ces´aro ..........................................
§ XII.27 Inequalities on powers and factorials related to the gamma
function ....................................................................
§ XII.28 Arithmetical products involving the gamma function .............
§ XII.29 Monotonicity and convexity results of certain expressions of
gamma function ...........................................................
§ XII.30 Left factorial function ...................................................

447

§

Chapter XIII
ESTIMATES INVOLVING FINITE GROUPS AND SEMI-SIMPLE RINGS ....
§ XIII. 1 Maximal order of an element in the symmetric group ............
§ XIII. 2 A sum on the order of elements of Sn ................................

§ XIII. 3 Statistical problems in Sn ...............................................
§ XIII. 4 Probability of generating the symmetric group .....................
§ XIII. 5 Primitive subgroups of Sn ..............................................
§ XIII. 6 Number of solutions of x k = 1 in symmetric groups ..............
§ XIII. 7 On the dimensions of representations of Sn .........................
§ XIII. 8 Conjugacy classes of the alternating group of degree n ...........
§ XIII. 9 An estimate for the order of rational matrices ......................
§ XIII.10 On kth power coset representatives mod p ..........................
§ XIII.11 Arithmetical properties of permutations of integers ..............
§ XIII.12 Number of non-isomorphic abelian groups of order n ...........
§ XIII.13 Abelian groups of a given order ......................................
§ XIII.14 Number of non-isomorphic abelian groups in short intervals ...
§ XIII.15 Number of representations of n as a product of k-full
numbers ...................................................................
§ XIII.16 Number of distinct values taken by a(n) and related
problems ..................................................................
§ XIII.17 Number of n ≤ x with a(n) = a(n + 1). The functions
a(n) at consecutive integers ...........................................
§ XIII.18 Sums involving ( (n + 1) − ␻(n + 1)) · a(n),
d(n + 1) a(n), ␻(n + 1) a(n) ..........................................
1
1
§ XIII.19 On sums involving
and
..............................
a(n)
log a(n)
§ XIII.20 The iterates of a(n) ......................................................
§ XIII.21 Statistical theorems on the embedding of abelian groups into
symmetrical ones ........................................................

§ XIII.22 Probabilistic results in group theory .................................
§ XIII.23 Finite abelian group cohesion .........................................
§ XIII.24 Number of non-isomorphic groups of order n .....................
§ XIII.25 Density of finite simple group orders ................................

448
450
451
451
452
457
459
459
460
461
462
463
464
465
466
467
467
467
468
472
472
473
474
475
476

477
477
478
479
480
481
483


Table of Contents
§

XIII.26
XIII.27
§ XIII.28
§ XIII.29
§

§

XIII.30
§ XIII.31
§ XIII.32
§

XIII.33
§ XIII.34

xxi

Large cyclic subgroups of finite groups .............................
Counting solvable, cyclic, nilpotent groups orders ................
On C-groups ..............................................................
The order of directly indecomposable groups. Direct factors
of a finite abelian groups ...............................................
On a family of almost cyclic finite groups ..........................
Asymptotic results for elements of a semigroup ..................
Number of non-isomorphic semi-simple finite rings of order
n ............................................................................
On a problem of Rohrbach for finite groups .......................
On cocyclity of finite groups ..........................................

Chapter XIV
PARTITIONS ...................................................................................
§ XIV. 1 Unrestricted partitions of an integer ..................................
§ XIV. 2 Partitions of n into exactly k positive parts ..........................
§ XIV. 3 Partitions of n into at most k summands .............................
§ XIV. 4 Unequal partitions of n containing each a j as a summand .......
§ XIV. 5 Partitions of n into members of a finite set ..........................
§ XIV. 6 Partitions of n without a given subsum ...............................
§ XIV. 7 Partitions of n which no part is repeated more than t times ......
§ XIV. 8 Partitions of n whose parts are ≥ m ..................................
§ XIV. 9 Partitions of n into unequal parts ≥ m ...............................
§ XIV.10 On the subsums of a partition .........................................
§ XIV.11 On other subsums of a partition .......................................
§ XIV.12 Partitions of j-partite numbers into k summands ..................
§ XIV.13 On a result of Tur´an .....................................................
§ XIV.14 Statistical theory of partitions .........................................
§ XIV.15 Partitions of n into distinct parts all ≡ ai (mod m) ................
§ XIV.16 Partitions with congruences conditions ..............................

§ XIV.17 Partitions of n whose parts are relatively prime, or prime
to n, etc. ...................................................................
§ XIV.18 Partitions of n whose parts ai (i = 1, k) satisfy
a1 |a2 | . . . |ak ..............................................................
§ XIV.19 Partitions of n as sums of powers of 2 ...............................
§ XIV.20 Partitions of n into powers of r (≥ 2) ................................
§ XIV.21 On a problem of Frobenius ............................................
§ XIV.22 An Abel-Tauber problem for partitions ..............................
§ XIV.23 On partitions of the positive integers with no x, y, z
belonging to distinct classes satisfying x + y = z ................
§ XIV.24 On certain partitions of n into r ≥ 2 distinct pairs ................
§ XIV.25 Additively independent partitions ....................................
§ XIV.26 A problem in “factorisatio numerorum” of Kalm´ar ...............

484
484
485
486
487
488
489
490
490

491
491
493
495
497
498

498
499
499
501
502
504
505
507
507
508
508
509
510
512
512
513
514
515
515
516
516


xxii

Table of Contents
§

XIV.27
XIV.28

§ XIV.29
§ XIV.30

Cyclotomic partitions ...................................................
Multiplicative properties of the partition function .................
Partitions into primes ...................................................
Partitions of N into terms of 1, 2, . . . , n, repeating a term
at most p times ...........................................................
§ XIV.31 Partition which assumes all integral values .........................
§ XIV.32 Partitions free of small summands ...................................
§

Chapter XV
CONGRUENCES, RESIDUES AND PRIMITIVE ROOTS ..........................
§ XV. 1 Addition of residue classes mod p .....................................
§ XV. 2 Residues of n n .............................................................
§ XV. 3 Distribution of quadratic nonresidues .................................
§ XV. 4 Distribution of quadratic residues ......................................
§ XV. 5 Sequences of consecutive quadratic nonresidues ....................
§ XV. 6 On residue difference sets ...............................................
§ XV. 7 Sets which contain a quadratic residue mod p for almost all
p ..............................................................................
§ XV. 8 Least prime quadratic residue ...........................................
§ XV. 9 Quadratic residues of squarefree integers .............................
§ XV.10 Least k-th power nonresidue ............................................
§ XV.11 Quadratic residues in arithmetic progressions .......................
§ XV.12 Bounds on n-th power residues (mod p) ..............................
§ XV.13 Positive d-th power residues ≤ x, with d|( p − 1), which
are prime to A .............................................................
§ XV.14 Distribution of r -th powers in a finite field ..........................

§ XV.15 P´
olya-Vinogradov inequality for quadratic characters .............
§ XV.16 Distribution questions concerning the Legendre symbol ..........
n
§ XV.17 A sum on
· n k ......................................................
p
§ XV.18 An exponential polynomial formed with the Legendre
symbol ......................................................................
§ XV.19 A mean value of a quadratic character sum ..........................
§ XV.20 Two sums involving Legendre’s symbol with primes ..............
§ XV.21 Least primitive roots mod p. Least primitive roots mod p 2 .
Number of solutions of congruence x n−1 ≡ 1(mod n) for n
composite ..................................................................
§ XV.22 Distribution of primitive roots of a prime ............................
§ XV.23 Artin’s conjecture on primitive roots ..................................
§ XV.24 Number of primitive roots ≤ x which are ≡ 1(mod k) ............
§ XV.25 Number of squarefull (squarefree) primitive roots ≤ x ...........
§ XV.26 Number of integers in [M + 1, M + N ] which are not
primitive roots (mod p) for any p ≤ N 1/2 ...........................
§ XV.27 Least prime primitive roots .............................................

519
520
520
520
521
521
523
523

524
524
526
528
529
530
530
530
531
532
534
534
534
535
535
536
537
537
537

538
541
542
543
543
544
544


Table of Contents

§

XV.28
XV.29
§ XV.30
§ XV.31
§ XV.32
§ XV.33
§ XV.34
§ XV.35
§ XV.36
§ XV.37
§ XV.38
§ XV.39
§ XV.40
§ XV.41
§ XV.42
§

xxiii

Fibonacci primitive roots ................................................
Distribution of primitive roots in finite fields ........................
Number of solutions to f (x) ≡ 0(mod m) counted mod m ......
Estimates on Legendre symbols of polynomials ....................
Number of solutions to f (x) ≡ a(mod p b ) ( p prime) .............
Number of residue classes k(mod r ) with f (k) ≡ 0(mod r ) .....
Zeros of polynomials over finite fields ...............................
Congruences on homogenous linear forms ..........................
Waring’s problem (mod p) ..............................................

Estimate of Mordell on congruences ..................................
Distribution of solutions of congruences .............................
On a set of congruences related to character sums .................
Small zeros of quadratic congruences mod p .......................
Congruence-preserving arithmetical functions ......................
On a congruence of Mirimanoff type .................................

Chapter XVI
ADDITIVE AND MULTIPLICATIVE FUNCTIONS..................................
§ XVI. 1 Erd˝
os’ theorem on additive functions with difference tending
to zero, generalizations, extensions and related results ..........
§ XVI. 2 Completely additive functions with restricted growth ............
§ XVI. 3 Tur´an-Kubilius inequality ..............................................
§ XVI. 4 Erd˝
os-Kac theorem ......................................................
§ XVI. 5 Erd˝
os-Wintner theorem .................................................
§ XVI. 6 Value distribution of differences of additive functions ............
§ XVI. 7 Erd˝
os-Wintner theorem for normed semigroups ...................
§ XVI. 8 Tur´an-Kubilius inequality and the Erd˝
os-Wintner theorem
for additive functions of a rational argument .......................
§ XVI. 9 Limit theorem for additive functions on ordered semigroups ....
§ XVI.10 Laws of iterated logarithm for additive functions .................
§ XVI.11 Limit laws and moments of additive functions in short
intervals ...................................................................
§ XVI.12 Distribution function of the sum of an additive and
multiplicative function .................................................

§ XVI.13 Moments and concentration of additive functions ................
§ XVI.14 Local theorems for additive functions ...............................
§ XVI.15 Additive functions on arithmetic progressions .....................
§ XVI.16 On differences of additive functions .................................
§ XVI.17 Prime-independent additive functions ...............................
§ XVI.18 Moments and Ces`aro means of additive functions ................
§ XVI.19 Minimax-theorem for additive functions ...........................
§ XVI.20 Maximal value of additive functions in short intervals ...........
§ XVI.21 Normal order of additive functions on sets of shifted primes ...

545
545
545
547
548
549
550
552
553
553
554
555
555
556
556

557
557
560
561

563
564
566
567
567
568
569
570
571
571
572
574
575
577
577
579
580
581


xxiv

Table of Contents
§

XVI.22
XVI.23
§ XVI.24
§ XVI.25
§ XVI.26

§

§

XVI.27

§

XVI.28
§ XVI.29
§ XVI.30
§ XVI.31
§ XVI.32
§ XVI.33
§ XVI.34
§ XVI.35
§ XVI.36
§

XVI.37

§

XVI.38

§

XVI.39

§

XVI.40

§

XVI.41

§

XVI.42
§ XVI.43
§ XVI.44
§ XVI.45
§ XVI.46
§ XVI.47
§ XVI.48
§ XVI.49
§

XVI.50

Uniformly distributed (mod 1) additive functions ................
Additive functions and almost periodicity ..........................
Characterization of multiplicative functions .......................
Multiplicative functions with small increments ...................
Conditions on a multiplicative function to be completely
multiplicative ............................................................
Delange’s theorem on mean-values of multiplicative
functions ..................................................................
Hal´asz’ theorem .........................................................

Wirsing’s theorem .......................................................
Mean value of f g and f ∗ g ..........................................
Mean value of f (P(n)), P a polynomial ...........................
Multiplicative functions | f | ≤ 1: Summation formulas .........
Indlekofer’s theorem ....................................................
Ces`aro means of additive functions ..................................
Multiplicative functions on short intervals .........................
Multiplicative functions on arithmetic progressions. Elliott’s
theorems ..................................................................
Effective mean value estimate for complex multiplicative
functions ..................................................................
A theorem of Levin, Timofeev and Tuliagonov on the
distribution of multiplicative functions. The
Bakshtys-Galambos theorems ........................................
Sums on multiplicative functions satisfying certain
conditions .................................................................
An asymptotic summation formula for multiplicative
functions with | f (n)| ≤ 1 .............................................
An -estimate for the remainder of sums of multiplicative
functions ..................................................................
The distribution of values of some multiplicative functions ....
Multiplicative functions and small divisors ........................
An estimate for submultiplicative functions .......................
Divisibility properties of some multiplicative functions .........
On multiplicative functions satisfying a congruence relation ...
Exponential sums with multiplicative function coefficients .....
Ramanujan expansions of multiplicative functions ...............
Asymptotic formulae for reciprocals of quotients of additive
and multiplicative functions ...........................................
Semigroup-valued multiplicative functions ........................

582
582
582
583
584
584
587
588
590
591
591
592
593
594
595
597

599
600
601
601
602
603
604
604
605
605
606
606

609

INDEX OF AUTHORS....................................................................... 611


PREFACE
It is the aim of this book to systematize and to present in an easily accessible framework
the most important results from some parts of Number Theory, which are expressed by inequalities
or by estimates.
This book focuses on the most important arithmetic functions in Number Theory, such as
n
ϕ(n), σ(n), d(n), ω(n), (n), µ(n), π(n), P(n), ψ(x, y), e(␣), ␹ (n),
, P(n, k) and so on, tok
gether with various generalizations, analogues and extensions of these functions, and also properties of some functions related to the distribution of the primes and of the quadratic residues and
to partitions, etc. It is sufficient to take a look at the contents in order to realize the variety of
the approached subjects in each chapter. The chapters are divided in consecutive “themes.” Each
theme expresses properties which are similar or contiguous by their nature.
We have attempted to make a selection which reflects the current situation in the domain
regarded. On the other hand, as a basic characteristic of this book, we have included the results
of the pioneers in the domains regarded, as well as some results reflecting the evolution from the
pioneer works up to recent ones. Our aim was to give the most precise references, i.e. original
ones, even when the results are standard and can be found in textbooks. To this purpose we have
used a wealth of literature, consisting of books, monographs, journals, separates, reviews from
Mathematical Reviews and from Zentralblatt f¨ur Mathematik, etc. Consequently, we hope that
our book will also be useful for the nonspecialist, who – if need be – can find the result or the
reference he needs. First of all, we consider the professional mathematician who works in a certain
domain of Number Theory and who wishes to use material outside his own field in Number
Theory. In this way, we hope to contribute to the unity of Number Theory despite of its great
variety.
Of course, the choice of subjects reflects the personality of the authors. Therefore,

we do not exclude the possibility that some important themes and aspects – even with respect to our proclaimed goal – are missing. We will be grateful to all readers who will honour us with their remarks. Their opinions will be considered with the greatest attention by the
authors.
Our book is not the first of this kind. The Handbook of Estimates in the Theory of Numbers
by B. Spairman and K.S. Williams (Carleton University, Ottawa) appeared in 1975. The book by
D.S. Mitrinovi´c and M.S. Popadi´c, Inequalities in Number Theory (Nauˇcni Podmladak, Univerzitet
u Niˇsu) appeared in 1978.
The latter monograph served as impulse for the present book, as Prof. D.S. Mitrinovi´c
had the intention to publish a second edition – revised and enlarged – of the monograph written
together with the late M.S. Popadi´c. Because of M.S. Popadi´c’s death, this project could not
be accomplished. Prof. D.S. Mitrinovi´c then addressed the invitation for cooperation to Prof.
J. S´andor. This circumstance led to an essentially new book, in concept, as well as in material.
Prof. D.S. Mitrinovi´c wishes to thank all mathematicians who have made remarks concerning
his previous book. These remarks have been taken into account if they refer to the material


xxvi

Preface

included in the present book. Prof. J. S´andor wishes to thank the mathematicians all over the world
who have had the kindness to offer him their papers. The gratefulness of J. S´andor is especially
addressed to the colleagues from the Mathematics Institute of Budapest (Hungary) as well as
from Institutul Matematic al Academiei Romˆane – Bucharest (Romania). The authors hope that
the mathematicians who have been in touch with them, in matters concerning the material of this
book, will recognise themselves in the above acknowledgements. The list would be too long to
mention them all.
The gratefulness of the authors is addressed to the staff of Kluwer Academic Publishers, especially to Dr. Paul Roos, Ms. Angelique Hempel and Ms. Anneke Pot for support while
typesetting the manuscript.
The camera-ready manuscript for the present book was prepared by Mr. Antonius Stanciu
(Timi¸soara, Romania) to whom the authors express their gratitude. The authors also acknowledge

the assistance of Mr. Dan Magiaru in the final elaboration of the text.

The Authors

Unfortunately, after the manuscript was finished and during its preparation for printing,
Professor D.S. Mitrinovi´c died (the 2nd of April, 1995), not having the chance to see his last work
in libraries.

June 1995

B.C.
J.S.


BASIC SYMBOLS
Below appear the most important symbols. The other ones are explained in the text.
f (x) = O(g(x)) or
f (x)
g(x)

For a range of x-values, there is a constant A
such that the inequality
| f (x)| ≤ A g(x)
holds over the range

f (x)

g(x)

f (x) = o(g(x))

g(x)

f (x), (or g(x) = O( f (x)))

as x → ∞, means
f (x)
=0
g(x)
(g(x) = 0 for x large.) The same meaning is
used when x → ∞ is replaced by x → ␣, for
any fixed ␣.
lim

x→∞

f (x) ∼ g(x)

as x → ∞, means
f (x)
=1
g(x)
(g(x) = 0 for x large.) The same is true when
x → ∞ is replaced with x → ␣.
lim

x→∞

f (x) =

(g(x))

f (x) = o(g(x)) does not hold.

f (x) =

+ (g(x))

There exists a positive constant K such that
f (x) > K g(x) is satisfied by values of x
surpassing all limit.

f (x) =

(g(x))

f (x) < −K g(x) is satisfied by values of x
surpassing all limit.

f (x) =

± (g(x))

we have both f (x) =
f (x) = (g(x))

+ (g(x))

and