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"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 97
Editorial Board
B. Bollobas, W . Fulton, A. Katok, F . Kirwan, P . Sarnak, B. Simon, B. T otaro
MUL TIPLICA TIVE NUMBER THEOR Y I:
CLASSICAL THEOR Y
Prime numbers are the multiplicative building blocks of natural numbers. Un-
derstanding their overall influence and especially... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
CAMBRIDGE STUDIES IN ADV ANCED MA THEMA TICS
All the titles listed below can be obtained from good booksellers of from Cambridge University
Press. For a complete series listing visit:
http://www .cambridge.org/series/sSeries.asp?code=CSAM
Already published
70 R. Iorio & V . Iorio F ourier analysis and partial different... | {
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Multiplicative Number Theory
I. Classical Theory
HUGH L. MONTGOMERY
University of Michigan, Ann Arbor
ROBERT C. VAUGHAN
P ennsylvania State University, University P ark | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
First published in print format
isbn-13 978-0-521-84903-6
isbn-13 978-0-511-25746-9
© Cambridge University Press 2006
2006
Information on this title... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
Dedicated to our teachers :
P . T . Bateman
J. H. H. Chalk
H. Davenport
T . Estermann
H. Halberstam
A. E. Ingham | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
T alet ¨ar t ¨ankandets b ¨orjan och slut.
Med tanken f ¨oddes talet.
Ut ¨ofver talet n˚ ar tanken icke.
Numbers are the beginning and end of thinking.
With thoughts were numbers born.
Beyond numbers thought does not reach.
Magnus Gustaf Mittag-Leffler, 1903 | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
Contents
Preface page xi
List of notation xiii
1 Dirichlet series: I 1
1.1 Generating functions and asymptotics 1
1.2 Analytic properties of Dirichlet series 11
1.3 Euler products and the zeta function 19
1.4 Notes 31
1.5 References 33
2 The elementary theory of arithmetic functions 35
2.1 Mean values 35
2.2 The prime ... | {
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viii Contents
4.4 Notes 133
4.5 References 134
5 Dirichlet series: II 137
5.1 The inverse Mellin transform 137
5.2 Summability 147
5.3 Notes 162
5.4 References 164
6 The Prime Number Theorem 168
6.1 A zero-free region 168
6.2 The Prime Number Theorem 179
6.3 Notes 192
6.4 References 195
7 Applications of the Prime Numb... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
Contents ix
11 Primes in arithmetic progressions: II 358
11.1 A zero-free region 358
11.2 Exceptional zeros 367
11.3 The Prime Number Theorem for arithmetic
progressions 377
11.4 Applications 386
11.5 Notes 391
11.6 References 393
12 Explicit formulæ 397
12.1 Classical formulæ 397
12.2 W eil’s explicit formula 410
12.3... | {
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x Contents
C The gamma function 520
C.1 Notes 531
C.2 References 533
D T opics in harmonic analysis 535
D.1 Pointwise convergence of Fourier series 535
D.2 The Poisson summation formula 538
D.3 Notes 542
D.4 References 542
Name index 544
Subject index 550 | {
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Preface
Our object is to introduce the interested student to the techniques, results, and
terminology of multiplicative number theory . It is not intended that our discus-
sion will always reach the research frontier. Rather, it is hoped that the material
here will prepare the student for intelligent reading of the mor... | {
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"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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xii Preface
twenty-five years in preparation—we hope to be a little quicker with the second
volume.
Many people have assisted us in this work—including P . T . Bateman, E.
Bombieri, T . Chan, J. B. Conrey , H. G. Diamond, T . Estermann, J. B. Friedlan-
der, S. W . Graham, S. M. Gonek, A. Granville, D. R. Heath-Brown, H.... | {
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Notation
Symbol Meaning Found on page
CThe set of complex numbers. 109
F p A field of p elements. 9
N The set of natural numbers, 1, 2, ... 114
Q The set of rational numbers. 120
R The set of real numbers. 43
TR /Z, known as the circle group or
the one-dimensional torus , which is
to say the real numbers modulo 1.
110
Z... | {
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xiv List of notation
Symbol Meaning Found on page
Ek The Euler numbers , also known as
the secant coefficients .
506
e(θ) = e2πi θ; the complex exponential
with period 1.
64, 108ff
L (s,χ ) A Dirichlet L -function. 120
Li(x ) =
∫x
0
du
log u with the Cauchy
principal value taken at 1; the
logarithmic integral.
189
li(x ... | {
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List of notation xv
Symbol Meaning Found on page
Ŵ(s,a) =
∫∞
a e−wws−1 d w; the incomplete
Gamma function .
327
γ The imaginary part of a zero of the
zeta function or of anL -function.
172
/Delta1 N (θ) = 1 + 2 ∑ N −1
n=1 (1 − n/N ) cos 2 πnθ;
known as the Fe j´er kernel .
174
ε(χ) = τ(χ)/
(
i κq 1/2 )
. 332
ζ(s) = ∑ ∞... | {
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xvi List of notation
Symbol Meaning Found on page
/Pi1(x ) = ∑
n≤x /Lambda1 (n)/log n. 416
π(x ) The number of primes not exceeding x .3
π(x ; q ,a) The number of p ≤ x such that p ≡ a
(mod q ),.
90, 358
π(x ,χ ) = ∑
p≤x χ( p). 377ff
ρ = β + i γ; a zero of the zeta function or
of an L -function.
173
ρ(u) The Dickman fu... | {
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List of notation xvii
Symbol Meaning Found on page
[x ] The unique integer such that
[x ] ≤ x < [x ] + 1; called the integer
part of x .
15, 24
{x }= x − [x ]; called the fractional part of x .2 4
∥ x ∥ The distance from x to the nearest
integer.
477
f (x ) = O (g(x )) | f (x )|≤ Cg (x ) where C is an absolute
constant... | {
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1
Dirichlet series: I
1.1 Generating functions and asymptotics
The general rationale of analytic number theory is to derive statistical informa-
tion about a sequence{an } from the analytic behaviour of an appropriate gen-
erating function, such as a power series ∑ an zn or a Dirichlet series ∑ an n−s .
The type of gen... | {
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2 Dirichlet series: I
coefficients of C (z) = A(z) B (z) are given by the formula
cn =
∑
k+m=n
ak bm . (1.1)
The terms are grouped according to the sum of the indices, because
zk zm = zk+m .
A Dirichlet series is a series of the form α(s) = ∑ ∞
n=1 an n−s where s is
a complex variable. If β(s) = ∑ ∞
m=1 bm m−s is a seco... | {
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"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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1.1 Generating functions and asymptotics 3
f (x ) ∼ g(x )( x →∞ ), if
lim
x →∞
f (x )
g(x ) = 1.
An instance of this arises in the formulation of the Prime Number Theorem
(PNT), which concerns the asymptotic size of the numberπ(x ) of prime num-
bers not exceeding x ; π(x ) = ∑
p≤x 1. Conjectured by Legendre in 1798, a... | {
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"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
4 Dirichlet series: I
0
0000
0000
0000
0000
00000 00000 00000 00000 1000000
Figure 1.1 Graph of π(x ) (solid) and x /log x (dotted) for 2 ≤ x ≤ 106 .
proved that π(x ) ≪ x /log x . This is of course weaker than the Prime Number
Theorem, but it was derived much earlier, in 1852. Chebyshev also showed
thatπ(x ) ≫ x /log ... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
1.1 Generating functions and asymptotics 5
size:
π(x ) = x
log x + x
(log x )2 + O
( x
(log x )3
)
.
This is also best possible, but the main term can be made still more elaborate to
give a smaller error term. Gauss was the first to propose a better approximation to
π(x ). Numerical studies led him to observe that the d... | {
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6 Dirichlet series: I
T able 1.1 V alues of π(x ), li(x),x /log x for x = 10k , 1 ≤ k ≤ 22.
x π(x ) li( x ) x /log x
10 4 5.12 4.34
102 25 29.08 21.71
103 168 176.56 144.76
104 1229 1245.09 1085.74
105 9592 9628.76 8685.89
106 78498 78626.50 72382.41
107 664579 664917.36 620420.69
108 5761455 5762208.33 5428681.02
109 ... | {
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1.1 Generating functions and asymptotics 7
That is, find constants a, b, ... , f so that the above is
a
(z − 1)3 + b
(z − 1)2 + c
z − 1 + d
z + 1 + e
z − ω + f
z − ω
where ω = e2πi /3 and ω = e−2πi /3 are the primitive cube roots of unity .
(c) Show that r (n) is the integer nearest ( n + 3)2 /12.
(d) Show that r (n) is... | {
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8 Dirichlet series: I
(b) Show that
∞∑
n=0
p(n; k)zn =
k∏
j =1
(1 − z j )−1
for |z| < 1.
(c) Show that
∞∑
n=0
p(n)zn =
∞∏
k=1
(1 − zk )−1
for |z| < 1.
(d) Show that
∞∑
n=0
pd (n)zn =
∞∏
k=1
(1 + zk )
for |z| < 1.
(e) Show that
∞∑
n=0
po (n)zn =
∞∏
k=1
(1 − z2k−1 )−1
for |z| < 1.
(f) By using the result of Exercise 2, o... | {
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1.1 Generating functions and asymptotics 9
(d) What needs to be said concerning the convergence of the series used
above?
6. (a) Let nk denote the total number of monic polynomials of degree k in
F p [x ]. Show that nk = pk .
(b) Let P1 ,P2 ,... be the irreducible monic polynomials in F p [x ], listed in
some (arbitrar... | {
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10 Dirichlet series: I
(j) Show that gn > 0 for all p and all n ≥ 1. (If P ∈ F p [x ] is irreducible and
has degree n, then the quotient ring F p [x ]/( P ) is a field of pn elements.
Thus we have proved that there is such a field, for each prime p and
integer n ≥ 1. It may be further shown that the order of a finite field... | {
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"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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1.2 Analytic properties of Dirichlet series 11
induction that
A(z) − B (z) =
∞∏
k=0
(
1 − z2k
)
for |z| < 1.
(e) Let the binary weight of n, denoted w(n), be the number of 1’s in the
binary expansion of n. That is, if n = 2k1 +···+ 2kr with k1 > ··· > kr ,
then w(n) = r . Show that A consists of those non-negative inte... | {
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12 Dirichlet series: I
T o see this we note that an = ( R(n − 1) − R(n)) ns0 , so that by partial
summation
N∑
n=M +1
an n−s =
N∑
n=M +1
( R(n − 1) − R(n))ns0 −s
= R( M ) M s0 −s−R( N ) N s0 −s −
N∑
n=M +1
R(n −1)((n −1)s0 −s − ns0 −s ).
The second factor in this last sum can be expressed as an integral,
(n − 1)s0 −s −... | {
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1.2 Analytic properties of Dirichlet series 13
W eierstrass that α(s) is analytic for σ>σ c , and that the differentiated series is
locally uniformly convergent to α′(s):
α′(s) =−
∞∑
n=1
an (log n)n−s (1.8)
for s in the half-plane σ>σ c .
Suppose that s0 is a point on the line of convergence (i.e., σ0 = σc ), and that
... | {
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14 Dirichlet series: I
Now suppose that σc ≥ 0. By Corollary 1.2 we know that the series in (1.10)
diverges when σ<σ c . Hence φ ≥ σc . T o complete the proof it suffices to show
that φ ≤ σc . Choose σ0 >σ c . By (1.7) with s = 0 and M = 0 we see that
A( N ) =− R( N ) N σ0 + σ0
∫ N
0
R(u)uσ0 −1 du .
Since R(u) is a boun... | {
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1.2 Analytic properties of Dirichlet series 15
Theorem 1.5 Suppose that α(s) = ∑ an n−s has abscissa of convergence σc .
If δ and ε are fixed, 0 <ε<δ< 1, then
α(s) ≪ τ1−δ+ε
uniformly for σ ≥ σc + δ. The implicit constant may depend on the coefficients
an ,o n δ, and on ε.
By the example found in Exercise 8 at the end of ... | {
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16 Dirichlet series: I
Suppose that f is analytic in a domain D, and that 0 ∈ D. Then f can
be expressed as a power series ∑ ∞
n=0 an zn in the disc |z| <r where r is the
distance from 0 to the boundary ∂D of D. Although Dirichlet series are analytic
functions, the situation regarding Dirichlet series expansions is ver... | {
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1.2 Analytic properties of Dirichlet series 17
for |s − 1| < 1 + δ′.I f s < 1 then all terms above are non-negative. Since
series of non-negative numbers may be arbitrarily rearranged, for −δ′ < s < 1
we may interchange the summations over k and n to see that
α(s) =
∞∑
n=1
an n−1
∞∑
k=0
(1 − s)k (log n)k
k!
=
∞∑
n=1
an... | {
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18 Dirichlet series: I
Deduce that the above is ≪ τN σc −σ+ε uniformly for s in the half-plane
σ ≥ σc + ε where the implicit constant may depend on ε and on the
sequence {an }.
(c) Show that
N∑
n=1
|an |n−σ = A( N ) N −σ+σc + (σ − σc )
∫ N
1
A(u)u−σ+σc −1 du
for any σ. Deduce that the above is ≪ N σa −σ+ε uniformly for... | {
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1.3 Euler products and the zeta function 19
(b) Show that if tr ≤ x < 2tr for some r , then A(x ) = [x ]it r where A(x ) =∑
n≤x an .
(c) Show that A(x ) ≪ 1 uniformly for x ≥ 1.
(d) Deduce that α(s) converges for σ> 0.
(e) Show that α(it ) does not converge; conclude that σc = 0.
(f) Show that if σ> 0, then
α(s) =
R∑
r... | {
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20 Dirichlet series: I
A function is called an arithmetic function if its domain is the set Z of inte-
gers, or some subset of the integers such as the natural numbers. An arithmetic
functionf (n) is said to be multiplicative if f (1) = 1 and if f (mn ) = f (m) f (n)
whenever ( m,n) = 1. Also, an arithmetic function f ... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
1.3 Euler products and the zeta function 21
so each sum on the right-hand side of (1.14) is absolutely convergent. Let
y be a positive real number, and let N be the set of those positive integers
composed entirely of primes not exceeding y, N ={ n : p|n ⇒ p ≤ y}. (Note
that 1 ∈ N.) Since a product of finitely many absol... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
22 Dirichlet series: I
determine the Dirichlet series generating functions of λ(n) and of µ(n) in terms
of the Riemann zeta function.
Corollary 1.10Fo r σ> 1,
∞∑
n=1
n−s = ζ(s) =
∏
p
(1 − p−s )−1 , (1.17)
∞∑
n=1
µ(n)n−s = 1
ζ(s) =
∏
p
(1 − p−s ), (1.18)
and
∞∑
n=1
λ(n)n−s = ζ(2s)
ζ(s) =
∏
p
(1 + p−s )−1 . (1.19)
Proof ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
1.3 Euler products and the zeta function 23
in view of (1.3), (1.17), (1.18), and Theorem 1.6. More generally , if
F (n) =
∑
d |n
f (d ) (1.21)
for all n, then, apart from questions of convergence,
∑
F (n)n−s = ζ(s)
∑
f (n)n−s .
By M ¨ obius inversion, the identity (1.21) is equivalent to the relation
f (n) =
∑
d |n
µ(... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
24 Dirichlet series: I
The important elementary identity
∑
d |n
/Lambda1 (d ) = log n (1.22)
is reflected in the relation
ζ(s)
(
− ζ′
ζ (s)
)
=− ζ′(s),
since
−ζ′(s) =
∞∑
n=1
(log n)n−s
for σ> 1.
W e now continue the zeta function beyond the half-plane in which it was
initially defined.
Theorem 1.12Suppose that σ> 0,x > 0... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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1.3 Euler products and the zeta function 25
–10
–
–
–
–
0
10
1 5
Figure 1.2 The Riemann zeta function ζ(s) for 0 < s ≤ 5.
By taking x = 1 in (1.23) we obtain in particular the identity
ζ(s) = s
s − 1 − s
∫ ∞
1
{u}u−s−1 du (1.24)
for σ> 0. Hence we have
Corollary 1.13 The Riemann zeta function has a simple pole at s = 1... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
26 Dirichlet series: I
W e now put the parameter x in (1.23) to good use.
Corollary 1.15 Let δ be fixed, δ> 0. Then for σ ≥ δ,s ̸=1,
∑
n≤x
n−s = x 1−s
1 − s + ζ(s) + O (τx −σ). (1.25)
In addition,
∑
n≤x
1
n = log x + C0 + O (1/x ) (1.26)
where C 0 is Euler’s constant,
C0 = 1 −
∫ ∞
1
{u}u−2 du = 0.5772156649 .... (1.27)
... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
1.3 Euler products and the zeta function 27
uniformly for s in the rectangle δ ≤ σ ≤ 2, |t |≤ 1, and
ζ(s) ≪ (1 + τ1−σ) min
( 1
|σ − 1| , log τ
)
uniformly for δ ≤ σ ≤ 2, |t |≥ 1.
Proof The first assertion is clear from (1.24). When |t | is larger, we obtain
a bound for |ζ(s)| by estimating the sum in (1.25). Assume that... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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28 Dirichlet series: I
6. Let σa (n) = ∑
d |n d a . Show that
∞∑
n=1
σa (n)σb (n)n−s = ζ(s)ζ(s − a)ζ(s − b)ζ(s − a − b)/ζ(2s − a − b)
when σ> max (1,1 +ℜ a,1 +ℜ b,1 +ℜ (a + b)).
7. Let F (s) = ∑
p (log p) p−s , G (s) = ∑
p p−s for σ> 1. Show that in this
half-plane,
− ζ′
ζ (s) =
∞∑
k=1
F (ks ),
F (s) =−
∞∑
d =1
µ(d ) ζ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
1.3 Euler products and the zeta function 29
(c) (Shafer 1984) Show that
∞∑
n=1
(−1)n (log n)n−1 = C0 log 2 − 1
2 (log 2) 2 .
12. (Stieltjes 1885) Show that if k is a positive integer, then
∑
n≤x
(log n)k
n = (log x )k+1
k + 1 + Ck + Ok
((log x )k
x
)
for x ≥ 1 where
Ck =
∫ ∞
1
{u}(log u)k−1 (k − log u)u−2 du .
Show tha... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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30 Dirichlet series: I
(c) Show that the parameters can be chosen so that (1.30)–(1.32) hold, say
by taking Nk = exp(1/εk ) and tk = ε1/2
k with εk tending rapidly to 0.
14. Let t (n) = (−1)/Omega1 (n)−ω(n) ∏
p|n ( p − 1)−1 , and put T (s) = ∑
n t (n)n−s .
(a) Show that for σ> 0, T (s) has the absolutely convergent Eul... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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1.4 Notes 31
1.4 Notes
Section 1.1. For a brief introduction to the Hardy–Littlewood circle method,
including its application to W aring’s problem, see Davenport (2005). For a
comprehensive account of the method, see V aughan (1997). Other examples
of the fruitful use of generating functions are found in many sources, ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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32 Dirichlet series: I
(cf. Mordell 1958). The square of the Dirichlet series in Exercise 1.2.8 has ab-
scissa of convergence 1/2; this bears on the result of Exercise 2.1.9. Information
concerning the convergence of the product of two Dirichlet series is found in
Exercises 1.3.2, 2.1.9, 5.2.16, and in Hardy & Riesz (1... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
1.5 References 33
work on Dirichlet series with natural boundaries see Estermann (1928a,b) and
Kurokawa (1987).
1.5 References
Andrews, G. E. (1976). The Theory of P artitions , Reprint. Cambridge: Cambridge Uni-
versity Press (1998).
Bachmann, P . (1894). Zahlentheorie, II, Die analytische Zahlentheorie , Leipzig:
T e... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
34 Dirichlet series: I
Jensen, J. L. W . V . (1884). Om Rækkers Konvergens, Tidsskrift for Math. (5) 2, 63–72.
(1887). Sur la fonction ζ(s) de Riemann, Comptes Rendus Acad. Sci. Paris 104,
1156–1159.
Knuth, D. E. (1962). Euler’s constant to 1271 places, Math. Comp. 16, 275–281.
Kurokawa, N. (1987). On certain Euler pro... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2
The elementary theory of arithmetic functions
2.1 Mean values
W e say that an arithmetic function F (n) has a mean value c if
lim
N →∞
1
N
N∑
n=1
F (n) = c.
In this section we develop a simple method by which mean values can be shown
to exist in many interesting cases.
If two arithmetic functions f and F are related ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
36 The elementary theory of arithmetic functions
Since [ y] = y + O (1), this is
= x
∑
d ≤x
f (d )
d + O
(∑
d ≤x
| f (d )|
)
. (2.3)
Thus F has the mean value ∑ ∞
d=1 f (d )/d if this series converges and if∑
d ≤x | f (d )|= o(x ). This approach, though somewhat crude, often yields use-
ful results.
Theorem 2.1Let ϕ(n)... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.1 Mean values 37
This is a relation of the shape (2.1) where f (d ) = µ(
√
d )i f d is a perfect square,
and f (d ) = 0 otherwise. Hence by (2.3),
Q(x ) = x
∑
d 2 ≤x
µ(d )
d 2 + O
(∑
d 2 ≤x
1
)
.
The error term is ≪ x 1/2 , and the sum in the main term is treated as in the
preceding proof. □
W e note that the argumen... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
38 The elementary theory of arithmetic functions
pairs ( k,m) for which km ≤ x , m ≤ x /y, and the third term subtracts those ak bm
for which k ≤ y, m ≤ x /y, since these ( k,m) were included in both the previous
terms. The advantage of (2.9) over (2.7) is that the number of terms is reduced
(≪ y + x /y instead of ≪ x ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.1 Mean values 39
Dirichlet convolution for multiplication. This ring is called the ring of formal
Dirichlet series . Manipulations of arithmetic functions in this way correspond
to manipulations of Dirichlet series without regard to convergence. This is
analogous to the ring of formal power series, in which multiplic... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
40 The elementary theory of arithmetic functions
(b) Show that for any fixed integer k > 1
∑
n≤x
n∈Qk
1 = x
ζ(k) + O
(
x 1/k )
for x ≥ 1.
4. (cf. Evelyn & Linfoot 1930) Let N be a positive integer, and suppose that
P is square-free.
(a) Show that the number of residue classes n (mod P 2 ) for which ( n,P 2 )
is square-f... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.1 Mean values 41
for all n if and only if
f (n) =
∑
m
n|m
µ(m/n) F (m).
7. (Jarn´ ık 1926; cf. Bombieri & Pila 1989) Let C be a simple closed curve in
the plane, of arc length L . Show that the number of ‘lattice points’ ( m,n),
m,n ∈ Z, lying on C is at most L + 1. Show that if C is strictly convex
then the number o... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
42 The elementary theory of arithmetic functions
(c) Show that
∞∑
d =1
µ(d )2 log d
d ϕ(d ) =
(∑
p
log p
p2 − p + 1
)∏
p
(
1 + 1
p( p − 1)
)
.
(d) Show that for x ≥ 2,
∑
n≤x
1
ϕ(n) = ζ(2)ζ(3)
ζ(6)
(
log x +C0 −
∑
p
log p
p2 − p + 1
)
+ O ((log x )/x ).
14. Let κ be a fixed real number. Show that
∑
n≤x
(ϕ(n)
n
)κ
= c(κ)x... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.1 Mean values 43
(c) Show that for any real number x ≥ 2 and any positive integer q ,
∑
n≤x
(n,q )=1
1
ϕ(n) = ζ(2)ζ(3)
ζ(6)
∏
p|q
(
1 − p
p2 − p + 1
)(
log x + C0 +
∑
p|q
log p
p − 1
−
∑
p∤q
log p
p2 − p + 1
)
+ O
(
2ω(q ) log x
x
)
.
17. (cf. W ard 1927) Show that for x ≥ 2,
∑
n≤x
µ(n)2
ϕ(n) = log x + C0 +
∑
p
log p... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
44 The elementary theory of arithmetic functions
embarrassing that this is the best-known upper bound for gaps between
sums of two squares.)
22. (Feller & T ornier 1932) Let f (n) denote the multiplicative function such
that f ( p) = 1 for all p, and f ( pk ) =− 1 whenever k > 1.
(a) Show that
∞∑
n=1
f (n)
ns = ζ(s)
∏
... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.1 Mean values 45
23. Let B1 (x ) = x − 1/2, as in Appendix B.
(a) Show that
∑
n≤x
1
n = log x + C0 − B1 ({x })/x + O (1/x 2 ).
(b) Write ∑
n≤x d (n) = x log x + (2C0 − 1)x + /Delta1 (x ). Show that
/Delta1 (x ) =− 2
∑
n≤√x
B1 ({x /n}) + O (1).
(c) Show that
∫X
0 /Delta1 (x ) dx ≪ X .
(d) Deduce that
∑
n≤X
d (n)( X − ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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46 The elementary theory of arithmetic functions
(c) Suppose that Q ≥ 1 is an integer, B ≥ 1, and that 1 /Q3 ≤± f ′′(x ) ≤
B/Q3 for 0 ≤ x ≤ N where the choice of sign is independent of
x . Show that numbers ar , qr , Nr can be determined, 0 ≤ r ≤ R for
some R, so that (i) ( ar ,qr ) = 1, (ii) qr ≤ Q, (iii) | f ′( Nr ) ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.2 Estimates of Chebyshev and of Mertens 47
this sort, we shall assume without comment that the reader understands that it
suffices to prove the result for all sufficiently largex .
Proof By applying the M ¨ obius inversion formula to (1.22) we find that
/Lambda1 (n) =
∑
d |n
µ(d ) log n/d .
Thus by (2.7) it follows that... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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48 The elementary theory of arithmetic functions
and hope that
−
∑
d ∈D
ad log d
d is near 1 . (2.14)
By the definition of T (x ) we see that the left-hand side of (2.12) is
∑
dn ≤x
ad log n =
∑
dn ≤x
ad
∑
k|n
/Lambda1 (k) =
∑
dkm ≤x
ad /Lambda1 (k)
(2.15)
=
∑
k≤x
/Lambda1 (k) E (x /k)
where E ( y) = ∑
dm ≤y ad = ∑
d ad... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.2 Estimates of Chebyshev and of Mertens 49
and
ψ(x ) ≤ (1.1056)x + O ((log x )2 ).
By computing the implicit constants one can use this method to determine a
constantx0 such that ψ(2x ) − ψ(x ) > x /2 for all x > x0 . Since the contribution
of the proper prime powers is small, it follows that there is at least one pr... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
50 The elementary theory of arithmetic functions
equivalent to the estimates ϑ(x ) ∼ x , ψ(x ) ∼ x . By partial summation it is
easily seen that the PNT implies that
∑
p≤x
log p
p ∼ log x ,
and that
∑
p≤x
1
p ∼ log log x .
However, these assertions are weaker than PNT , as we can derive them from
Theorem 2.4.
Theorem 2... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.2 Estimates of Chebyshev and of Mertens 51
by Theorem 2.4. W e now prove (d) without determining the value of the con-
stantb. W e express (b) in the form L (x ) = log x + R(x ) where R(x ) ≪ 1.
Then
∑
p≤x
1
p =
∫ x
2−
(log u)−1 dL (u) =
∫ x
2−
1
log u d log u +
∫ x
2−
dR (u)
log u
=
∫ x
2−
du
u log u +
[ R(u)
log u
... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
52 The elementary theory of arithmetic functions
so that from (2.16) we have
∑
1<n≤x
/Lambda1 (n)
n log n = log log x + c + O (1/log x ).
By Corollary 1.15 this can be written
∑
1<n≤x
/Lambda1 (n)
n log n =
∑
n≤log x
1
n + (c − C0 ) + O (1/log 2 x ).
Since this is trivial when 1 ≤ x < 2, the above holds for all x ≥ 1. ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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2.2 Estimates of Chebyshev and of Mertens 53
Then there is an x0 such that ψ(x ) ≤ (a + ε)x for all x ≥ x0 , and hence
∫ x
1
ψ(u)u−2 du ≤
∫ x0
1
ψ(u)u−2 du +(a + ε)
∫ x
x0
u−1 du ≤ (a + ε) log x + Oε(1).
Since this holds for arbitrary ε> 0, it follows that
∫x
1 ψ(u)u−2 du ≤ (a +
o(1)) log x . Thus by Theorem 2.7(c) we ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
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54 The elementary theory of arithmetic functions
Thus for the coefficients an we have an analogue of Mertens’ esti-
mate of Theorem 2.7(b), but not an analogue of the Prime Number
Theorem.
4. (Golomb 1992) Let dx denote the least common multiple of the positive
integers not exceeding x . Show that
(2n
n
)
=
∞∏
k=1
d (−1... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.3 Applications to arithmetic functions 55
Theorem 2.9 F or all n ≥ 3,
ϕ(n) ≥ n
log log n
(
e−C0 + O (1/log log n)
)
,
and there are infinitely many n for which the above relation holds with equality.
ProofLet R be the set of those n for which ϕ(n)/n <ϕ (m)/m for all m < n.
W e first prove the inequality for these ‘reco... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
56 The elementary theory of arithmetic functions
can be large, but not nearly as large as √n. Indeed, for each ε> 0 there is a
constant C (ε) such that
d (n) ≤ C (ε)nε (2.20)
for all n ≥ 1. T o see this we express n in terms of its canonical factorization,
n = ∏
p pa , so that
d (n)
nε =
∏
p
a + 1
paε =
∏
p
f p (a),
sa... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.3 Applications to arithmetic functions 57
the ‘probability’ that d |n when n is ‘randomly chosen’ is 1 /d .I f d1 and d2
are two fixed numbers then the ‘probability’ that d1 |n and d2 |n is 1 /[d1 ,d2 ].
If ( d1 ,d2 ) = 1 then this ‘probability’ is 1 /(d1 d2 ), and we see that the ‘events’
d1 |n, d2 |n are ‘independen... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
58 The elementary theory of arithmetic functions
so that ω(n) = ∑
p X p (n). If we were to treat the X p as though they
were independent random variables then we would have E( X p ) = 1/p,
V ar(X p ) = (1 − 1/p)/p. Hence we expect that the average of ω(n) should be
approximately
E
(∑
p≤n
X p
)
=
∑
p≤n
E( X p ) =
∑
p≤n
... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.3 Applications to arithmetic functions 59
Corollary 2.13 (Hardy–Ramanujan) F or almost all n, ω(n) ∼ /Omega1 (n) ∼
log log n.
Note that in analytic number theory we say ‘almost all’ when the excep-
tional set has asymptotic density 0; this conflicts with the usage in some
parts of algebra, where the term means that th... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
60 The elementary theory of arithmetic functions
By the integral test the sum on the right is
=
∫ x
e
(log log x − log log u)2 du + O ((log log x )2 ).
By integrating by parts twice we find that this integral is
−e(log log x )2 −2e log log x +2
∫ x
2
1 + log log x −log log u
(log u)2 du ≪ x
(log x )2 .
Thus
(∑
1<n≤x
(ω(... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.3 Applications to arithmetic functions 61
for all x ≥ 1, and that
∑
pk
k≥2
f ( pk )k log p
pk ≤ A. (2.29)
Then for x ≥ 2,
∑
n≤x
f (n) ≪ ( A + 1) x
log x
∑
n≤x
f (n)
n .
W e note that this is sharper than the trivial estimate
∑
n≤x
f (n) ≤ x
∑
n≤x
f (n)/n (2.30)
that holds whenever f ≥ 0.
If f ≥ 0 and f is multiplicat... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
62 The elementary theory of arithmetic functions
Thus we see that ϕ(n)/n is not often very small.
Proof of Theorem 2.14 The desired bound is obtained by adding the two
estimates
∑
n≤x
f (n) log x
n ≪ x
∑
n≤x
f (n)
n , (2.33)
∑
n≤x
f (n) log n ≪ Ax
∑
n≤x
f (n)
n . (2.34)
The first of these is immediate, since f ≥ 0 and l... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.3 Applications to arithmetic functions 63
2. Show that d (n) ≤
√
3n with equality if and only if n = 12.
3. Let f (n) = ∏
p|n (1 + p−1/2 ).
(a) Show that there is a constant a such that if n ≥ 3, then
f (n) < exp
(
a(log n)1/2 (log log n)−1 )
.
(b) Show that ∑
n≤x f (n) = cx + O
(
x 1/2 )
where c = ∏
p (1 + p−3/2 ).
... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
64 The elementary theory of arithmetic functions
7. (Bateman 1949) Let /Phi1 q (z) denote the q th cyclotomic polynomial,
/Phi1 q (z) =
q∏
a=1
(a,q )=1
(z − e(a/q ))
where e(θ) = e2πi θ.
(a) Show that
∏
d |q
/Phi1 d (z) = zq − 1.
(b) Show that
/Phi1 q (z) =
∏
d |q
(zd − 1)µ(q /d ) .
(c) If P (z) = ∑ pn zn and Q(z) = ∑ ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.4 The distribution of /Omega1 (n) − ω(n)6 5
(e) Deduce that the second and third terms in (2.35) are ≪ 1.
(f) Conclude that
/Sigma1 2 = x (log log x )2 + (2b + 1) log log x + O (x )
where b is the constant in Theorem 2.7(d).
(g) Show that the left-hand side of (2.23) is = x log log x + O (x ).
(h) Show that the left-... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
66 The elementary theory of arithmetic functions
uniform in k. This is indeed the case, as we see from the following quantitative
form of R´ enyi’s theorem.
Theorem 2.16F or any non-negative integer k , and any x ≥ 2,
Nk (x ) = dk x + O
((3
4
)k
x 1/2 (log x )4/3 )
.
In preparation for the proof of this result we first ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.4 The distribution of /Omega1 (n) − ω(n)6 7
By Lemma 2.17 this is
6
π2 x
∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
1
f
∏
p| f
(1 + p−1 )−1 + O
⎛
⎜
⎜
⎜
⎜
⎝
x 1/2 ∑
f ≤x
f ∈F
/Omega1 ( f )−ω( f )=k
f −1/2 ∏
p| f
(
1 − p−1/2 )−1
⎞
⎟
⎟
⎟
⎟
⎠
.
In order to appreciate the nature of these sums it is helpful to observe that each
me... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
68 The elementary theory of arithmetic functions
Hence we have the stated result with
dk = 6
π2
∑
f ∈F
/Omega1 ( f )−ω( f )=k
1
f
∏
p| f
(
1 + 1
p
)−1
.
T o see that (2.36) holds, it suffices to multiply this by zk and sum over k. □
2.4.1 Exercise
1. Let dk be as in (2.36). Show that
dk = c2−k + O (5−k )
where
c = 1
4
∏... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
2.5 Notes 69
by van der Corput (1928), Chih (1950), Richert (1953), Kolesnik (1969, 1973,
1982, 1985), Iwaniec & Mozzochi (1988), and by Huxley (1993), who showed
that/Delta1 (x ) ≪ x 23/73+ε. In the opposite direction, Hardy (1916) showed that
/Delta1 (x ) = /Omega1 ±(x 1/4 ). Soundararajan (2003) showed that
/Delta1 ... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
70 The elementary theory of arithmetic functions
pp. 285–288). Polynomials can be found that produce better constants, but
Gorshkov (1956) showed that the supremum of such constants is< 1, so
the Prime Number Theorem cannot be established by this method. For more
on this subject, see Montgomery (1994, Chapter 10), Prit... | {
"creation_date": "2025-02-03",
"file_name": "(Cambridge Studies in Advanced Mathematics 97) Hugh L. Montgomery, Robert C. Vaughan - Multiplicative number theory I_ Classical theory-Cambridge University Press (2006).pdf",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
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