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82 Principles and first examples of sieve methods
3.2 The Selberg lambda-squared method
Let /Lambda1 n be a real-valued arithmetic function such that /Lambda1 1 = 1. Then
(∑
d |n
/Lambda1 d
)2
≥
{ 1i f n = 1,
0i f n > 1.
This simple observation can be used to obtain an upper bound for S(x ,y; P );
namely
S(x ,y; P ) ≤
∑... | {
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3.2 The Selberg lambda-squared method 83
where
L P (z) =
∑
n≤z
n| P
µ(n)2
ϕ(n) .
Proof Clearly we may assume that P is square-free. Since [ d ,e](d ,e) = de
and ∑
d |n ϕ(d ) = n, we see that
1
[d ,e] = (d ,e)
de = 1
de
∑
f |d ,f |e
ϕ( f ).
Hence
∑
d | P,e| P
/Lambda1 d /Lambda1 e
[d ,e] =
∑
f | P
ϕ( f )
∑
d
f |d | P
/L... | {
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84 Principles and first examples of sieve methods
Here the right-hand side is minimized by taking
yf = µ( f )
ϕ( f )L P (z) (3.15)
for f ≤ z, and we note that these yf satisfy (3.13). Hence the minimum of the
quadratic form in (3.10), subject to /Lambda1 1 = 1, is precisely 1 /L P (z); this gives the
main term.
W e now ... | {
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3.2 The Selberg lambda-squared method 85
Since s(m) ≤ m, this latter sum is
≥
∑
m≤z
1
m > log z.
Here the last inequality is obtained by the integral test. With more work one can
derive an asymptotic formula for the the sum in (3.18) (recall Exercise 2.1.17).
By taking z = y1/2 in Theorem 3.2, and appealing to (3.18), ... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
86 Principles and first examples of sieve methods
Proof Let
P1 =
∏
p| P
p≤√y
p, q1 =
∏
p∤ P
p≤√y
p.
Theorem 3.3 provides an upper bound for M ( y; q1 P1 ), and hence by Lemma
3.5 we have an upper bound for M ( y; P1 ). T o complete the argument it suffices
to note that S(x ,y; P ) ≤ S(x ,y; P1 ) ≤ M ( y; P1 ), and to app... | {
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3.2 The Selberg lambda-squared method 87
Then by Exercise 2.1.17 and Mertens’ estimates (Theorem 2.7) it follows that
this is1
4 (3 − 2 log 2) log y + O (1).
3.2.1 Exercises
1. Let /Lambda1 d be defined as in the proof of Theorem 3.2.
(a) Show that
/Lambda1 d ≪ d
L P (z)ϕ(d ) log 2z
d
for d ≤ z.
(b) Use the above to giv... | {
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88 Principles and first examples of sieve methods
5. (Hensley 1978)
(a) Let P = ∏
p≤√y p. Show that the number of n, x < n ≤ x + y, such
that /Omega1 (n) = 2, is
≤ S(x ,y; P ) +
∑
p≤√y
(
π
(x + y
p
)
− π
(x
p
))
.
(b) By using Theorem 3.3 and Corollary 3.4, show that for y ≥ 2,
∑
x <n≤x +y
/Omega1 (n)=2
1 ≤ 2 y log log ... | {
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3.3 Sifting an arithmetic progression 89
(b) Deduce that
∑
n≤z
(n,q )=1
µ(n)2
ϕ(n) ≥ ϕ(q )
q
∑
n≤z
µ(n)2
ϕ(n) .
10. (Hooley 1972; Montgomery & V aughan 1979)
(a) Let λ+
d be an upper bound sifting function such that λ+
d= 0 for all
d > z. Show that for any q ,
0 ≤ ϕ(q )
q
∑
d
(d ,q )=1
λ+
d
d ≤
∑
d
λ+
d
d .
(Hint: Mult... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
90 Principles and first examples of sieve methods
Proof By the Chinese remainder theorem there is a number c such that c ≡ a p
(mod p) for every p| P . Put n = m − c. Thus the inequality x < m ≤ x + y is
equivalent to x − c < n ≤ x − c + y, and the condition that p| P implies m ̸≡
a p (mod p) is equivalent to ( n,P ) = ... | {
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3.4 T win primes 91
Thus by Theorem 3.8, the number of primes p, x < p ≤ x + y, such that p ≡ a
(mod q ) and ( p,P ) = 1 satisfies the bound (3.23). T o complete the proof it
remains to note that the number of primes p, x < p ≤ x + y, such that p ≡ a
(mod q ) and p| P is at most ω( P ) ≤ √y/q , which can be absorbed in ... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
92 Principles and first examples of sieve methods
T o continue from this point, one should specify the choice of λm , and then
estimate the main term and error term. In the context of Selberg’s /Lambda1 2 method,
we have real /Lambda1 d with /Lambda1 1 and /Lambda1 d = 0 for d > z. The number of n ∈ (x ,x + y]
that surv... | {
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3.4 T win primes 93
The linear change of variables from /Lambda1 d to yf is invertible:
/Lambda1 d = d
b(d )
∑
f
d | f | P
yf µ( f /d ) . (3.30)
By the above formulæ we see that the condition that /Lambda1 d = 0 for d > z is
equivalent to the condition that yf = 0 for f > z. Also, the condition that
/Lambda1 1 = 1 is e... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
94 Principles and first examples of sieve methods
Proof W e first estimate L as given in (3.33). W e have b(2) = 1 and b( p) = 2
for p > 2. Since µ(m)2 g(m) is a multiplicative function that takes the value
2/( p − 2) when m = p > 2, and since d (n)/n is a multiplicative function that
takes the value 2 /p when n = p, we ... | {
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3.4 T win primes 95
It remains to bound the error term in (3.26). Since 0 ≤ b([d ,e]) ≤ b(d )b(e),
the error term is
≪
(∑
d ≤z
b(d )|/Lambda1 d |
)2
.
From (3.30) and (3.34) we see that
/Lambda1 d = d
b(d )L
∑
f ≤z
d | f
µ( f )g( f )µ( f /d ) = µ(d )dg (d )
b(d )L
∑
m≤z/d
(m,d )=1
µ(m)2 g(m) .
Hence
∑
d ≤z
b(d )|/Lambd... | {
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96 Principles and first examples of sieve methods
Proof The number of twin primes for which 2 k−1 < p ≤ 2k is ≪ 2k /k2 .
Hence the contribution of such primes to the sum in question is ≪ 1/k2 . But∑ 1/k2 < ∞, so we obtain the stated result. □
Let r be an even non-zero integer. T o bound the number of primes p for
which ... | {
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3.4 T win primes 97
Let b′
1 ( p) = b1 ( p) for p ̸=p′, b′
1 ( p′) = b1 ( p′) + 1. The left-hand side above
is ≤ M (x ,y; b′
1), which by the inductive hypothesis is
≤ M (x ,y; b2 ) p − b1 ( p′) − 1
p − b2 ( p′)
∏
p| P
p̸=p′
(p − b1 ( p)
p − b2 ( p)
)
.
Thus
M (x ,y; b1 ) ≤ M (x ,y; b2 )
∏
p| P
(p − b1 ( p)
p − b2 ( p)... | {
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98 Principles and first examples of sieve methods
Proof Let r (n) denote the number of solutions of n = p + 2k . By Cauchy’s
inequality ,
(∑
n≤x
r (n)
)2
≤ N (x )
∑
n≤x
r (n)2 .
Thus to complete the proof it suffices to show that
∑
n≤x
r (n) ≫ x (x ≥ 4), (3.38)
and that
∑
n≤x
r (n)2 ≪ x . (3.39)
The first of these estimat... | {
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3.4 T win primes 99
Put n = k − j . Thus 0 < n ≤ y. Let h2 (m) denote the order of 2 modulo m,
which is to say that h2 (m) is the least positive integer h such that 2 h ≡ 1
(mod m). W e note that m|(2n − 1) if and only if h2 (m)|n. The number of such
n,0 < n ≤ y,i s ≤ y/h2 (m). There are also ≤ y choices of j . Thus to... | {
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100 Principles and first examples of sieve methods
2. Show that the number of primes p ≤ 2n such that 2 n − p is prime is
≤ 8c
⎛
⎜
⎝
∏
p|n
p>2
p − 1
p − 2
⎞
⎟
⎠2n
(log 2 n)2
(
1 + O
(log log 4 n
log 2 n
))
where c is the constant in Theorem 3.10.
3. (Erd ˝ os 1940, Ricci 1954)
(a) Show that
∑
r ≤x
c(r ) = x + O (log x )... | {
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3.5 Notes 101
(i) T ake b = a + 1/8, and suppose that d ( p) ≥ a log p for all p > p0 .
Show that the estimates of (f) and (h) are inconsistent if a > 15/16.
Thus conclude that
lim inf
p→∞
d ( p)
log p ≤ 15
16 .
4. Let r (n) be defined as in the proof of Theorem 3.15. Show that
∑
n≤x
r (n) ∼ x
log 2 .
5. Let r (n) be de... | {
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102 Principles and first examples of sieve methods
to Selberg (1952a,b). The /Lambda12 method of Selberg (1947) provides only upper
bounds, but lower bounds can also be derived from it by using ideas of Buchstab
(1938).
In contrast to the elegance of the Selberg /Lambda12 method, the further study of
sieves leads us to ... | {
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3.5 Notes 103
Hooley (1994) has shown that quite sharp sieve bounds can be derived using
the interrupted inclusion–exclusion idea that Brun started with. This approach
has been developed further by Ford & Halberstam (2000). An exposition of
sieves based on these ideas is given by Bateman & Diamond (2004, Chapters 12,
1... | {
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104 Principles and first examples of sieve methods
any positive k. Presumably r (n) = o(log n), but for all we know there could be,
although it seems unlikely , infinitely many n such that n − 2k is prime whenever
0 < 2k < n. The number n = 105 has this property , and is probably the largest
such number. The best upper b... | {
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3.6 References 105
(1950). On integers of the form 2 k + p and some related problems, Summa Brasil.
Math. 2, 113–123.
(1951). On some problems of Bellman and a theorem of Romanoff, J. Chinese Math.
Soc. (N. S.) 1, 409–421.
Erd ˝ os, P . & Tur´ an, P . (1935). Ein zahlentheoretischer Satz, Mitt. F orsch. Inst. Math.
Mec... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
106 Principles and first examples of sieve methods
Jurkat, W . B. & Richert, H.-E. (1965). An improvement in Selberg’s sieve method, I,
Acta Arith. 11, 217–240.
Lehmer, D. H. (1955). The distribution of totatives, Canad. J. Math. 7, 347–357.
van Lint, J. H. & Richert, H.-E. (1964). ¨Uber die Summe ∑
n≦x
p(n)<y
µ2 (n)
ϕ(... | {
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3.6 References 107
Boston: Academic Press, pp. 467–484; Collected P apers, V ol. 1. Berlin: Springer-
V erlag, 1989, pp. 675–69.
(1991). Lectures on Sieves, Collected P apers , V ol. 2. Berlin: Springer-V erlag,
pp. 65–247.
Titchmarsh, E. C. (1930). A divisor problem, Rend. Circ. Math . Palermo 54, 414–429.
Tsang, K. M... | {
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4
Primes in arithmetic progressions: I
4.1 Additive characters
If f (z) = ∑ ∞
n=0 cn zn is a power series, we can restrict our attention to terms
for which n has prescribed parity by considering
1
2 f (z) + 1
2 f (−z) =
∞∑
n=0
n≡ 0 (2)
cn zn
or
1
2 f (z) − 1
2 f (−z) =
∞∑
n=0
n≡1 (2)
cn zn .
That is, we can express the... | {
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4.1 Additive characters 109
unless ζ = 1. Hence
1
q
q∑
k=1
e(−ka /q )e(kn /q ) =
{ 1i f n ≡ a (mod q ),
0 otherwise, (4.1)
and thus the characteristic function of an arithmetic progression (mod q ) can be
expressed as a linear combination of the sequences e(kn /q ). These functions
are called the additive characters (m... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
110 Primes in arithmetic progressions: I
and Fourier expansion of a function f ∈ L 1 (T), but the situation here is simpler
because our sums have only finitely many terms.
Let v (h) be the vector v (h) = [e(h/q ),e(2h/q ),..., e((q − 1)h/q ),1].
From (4.1) we see that two such vectors v (h1 ) and v (h2 ) are orthogonal ... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
4.1 Additive characters 111
Proof The first assertion is evident, as each term in the sum (4.5) has period
q . As for the second, suppose that q = q1 q2 where ( q1 ,q2 ) = 1. By the Chinese
Remainder Theorem, for each a (mod q ) there is a unique pair a1 ,a2 with ai
determined (mod qi ), so that a ≡ a1 q2 + a2 q1 (mod q... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
112 Primes in arithmetic progressions: I
Here the first sum is pk if pk |n, and is 0 otherwise. Similarly , the second
sum is pk−1 if pk−1 |n, and is 0 otherwise. Hence the above is
=
⎧
⎨
⎩
0i f pk−1 ∤ n,
− pk−1 if pk−1 ∥ n,
pk − pk−1 if pk |n
= µ
(
pk /(n, pk )
)
ϕ
(
pk /(n, pk )
)ϕ( pk ).
The general case of (4.7) now... | {
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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... |
4.1 Additive characters 113
3. Show that |cq (n)|≤ (q ,n).
4. (Carmichael 1932)
(a) Show that if q > 1, then
q∑
n=1
cq (n) = 0.
(b) Show that if q1 ̸=q2 and [ q1 ,q2 ]|N , then
N∑
n=1
cq1 (n)cq2 (n) = 0.
(c) Show that if q |N , then
N∑
n=1
cq (n)2 = N ϕ(q ).
5. (Grytczuk 1981; cf. Redmond 1983) Show that
∑
d |q
|cd (n)... | {
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114 Primes in arithmetic progressions: I
Let d be a fixed positive integer. Show that
∑
n≤x
d |n
F (n) = x
d
∞∑
r =1
f (r )
r (d ,r ) + o(x )
as x →∞ .
(b) Suppose that (4.10) holds. Show that
lim
x →∞
1
x
∑
n≤x
F (n)cq (n) = ϕ(q )
∞∑
r =1
q |r
f (r )
r .
(c) Put
aq =
∞∑
r =1
q |r
f (r )
r .
Show that if
∞∑
r =1
| f (r ... | {
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4.2 Dirichlet characters 115
4.2 Dirichlet characters
In the preceding section we expressed the characteristic function of an arithmetic
progression as a linear combination of additive characters. For purposes of
multiplicative number theory we shall similarly represent the characteristic
function of a reduced residue ... | {
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116 Primes in arithmetic progressions: I
Since the characters are now known explicitly , the remaining assertions are
easily verified.□
Next we describe the characters of the direct product of two groups in terms
of the characters of the factors.
Lemma 4.3Suppose that G 1 and G 2 are finite abelian groups, and that G =
G... | {
"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... |
4.2 Dirichlet characters 117
Corollary 4.5 The multiplicative group (Z/q Z)× of reduced residue classes
(mod q ) has ϕ(q ) Dirichlet characters. If χ is such a character , then
q∑
n=1
(n,q )=1
χ(n) =
{ ϕ(q ) if χ = χ0 ,
0 otherwise. (4.14)
If (n,q ) = 1, then
∑
χ
χ(n) =
{ ϕ(q ) if n ≡ 1 (mod q ),
0 otherwise, (4.15)
wh... | {
"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... |
118 Primes in arithmetic progressions: I
called the index of n, and is denoted ν = indg n. From Lemma 4.2 it follows
that the characters (mod pα), p > 2, are given by
χk (n) = e
(k indg n
ϕ( pα)
)
(4.16)
for ( n, p) = 1. W e obtain ϕ( pα) different characters by allowing k to assume
integral values in the range 1 ≤ k ≤... | {
"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... |
4.2 Dirichlet characters 119
consequently ( m,n + kq ) = 1. Then
f (mn ) = f (m(n + kq )) (by periodicity)
= f (m) f (n + kq ) (by multiplicativity)
= f (m) f (n) (by periodicity),
and the proof is complete. □
W e shall discuss further properties of Dirichlet characters in Chapter 9.
4.2.1 Exercises
1. Let G be a finite... | {
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120 Primes in arithmetic progressions: I
(b) Show that each kth root of unity occurs precisely ϕ(q )/k times among the
numbers χ(a)a s a runs over the ϕ(q ) reduced residue classes (mod q ).
6. Let χ be a character (mod q ) such that χ(a) =± 1 whenever ( a,q ) = 1, and
put S(χ) = ∑ q
n=1 nχ(n). Thus S(χ) is an integer.... | {
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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... |
4.3 Dirichlet L -functions 121
for σ> 1. Thus we see that
L (s,χ0 ) =
∞∑
n=1
(n,q )=1
n−s = ζ(s)
∏
p|q
(
1 − p−s )
(4.22)
for σ> 1. By (4.14) we see that if χ ̸=χ0 , then
∑
1≤n≤kq
χ(n) = 0
for k = 1,2,3,... . Hence
⏐
⏐
⏐
⏐
⏐
∑
n≤x
χ(n)
⏐
⏐
⏐
⏐
⏐≤ q (4.23)
for any x , so that by Theorem 1.3, the series (4.20) converges ... | {
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
122 Primes in arithmetic progressions: I
W e now use the identity (4.15) to capture a prescribed residue class. If
(a,q ) = 1, then
1
ϕ(q )
∑
χ
χ(a)χ(n) =
{ 1i f n ≡ a (mod q ),
0 otherwise (4.27)
where the sum is extended over all characters χ (mod q ). This is the multiplica-
tive analogue of (4.1). Hence if ( a,q ) ... | {
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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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4.3 Dirichlet L -functions 123
as s → 1+. Consequently
∞∑
n=1
n≡a (q )
/Lambda1 (n)
n =∞ .
Here the contribution of the proper prime powers is
∑
pk ≡a (q )
k≥2
log p
pk ≤
∑
p
log p
∞∑
k=2
p−k =
∑
p
log p
p( p − 1) < ∞, (4.31)
and thus we have
Corollary 4.10(Dirichlet’s theorem) If (a,q ) = 1, then there are infinitely
m... | {
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124 Primes in arithmetic progressions: I
L (s,χ ) = L (s,χ) by the Schwarz reflection principle, so that L (1,χ) = 0i f
L (1,χ ) = 0. Consequently L (1,χ ) ̸=0 for complex χ.
Case 2: Quadratic χ. Let r (n) = ∑
d |n χ(d ). Thus ∑ ∞
n=1 r (n)n−s =
ζ(s)L (s,χ ) for σ> 1, r (n) is multiplicative, and
r ( pα) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩... | {
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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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4.3 Dirichlet L -functions 125
Thus the error term above is
∑
n>x
χ(n) log n
n =
∫ ∞
x
log u
u dS (u)
=− S(x ) log x
x −
∫ ∞
x
S(u)(1 − log u)u−2 du
≪χ
log x
x .
As log n = ∑
d |n /Lambda1 (d ), the left-hand side of (4.33) is
∑
md ≤x
/Lambda1 (d )χ(md )
md =
∑
d ≤x
/Lambda1 (d )χ(d )
d
∑
m≤x /d
χ(m)
m . (4.34)
Here th... | {
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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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126 Primes in arithmetic progressions: I
(b)
∑
p≤x
n≡a (q )
log p
p = 1
ϕ(q ) log x + Oq (1),
(c)
∑
p≤x
n≡a (q )
1
p = 1
ϕ(q ) log log x + b(q ,a) + Oq
(1
log x
)
,
(d)
∏
p≤x
n≡a (q )
(
1 − 1
p
)−1
= c(q ,a)(log x )1/ϕ(q )
(
1 + Oq
(1
log x
))
where
b(q ,a) = 1
ϕ(q )
(
C0 +
∑
p|q
log
(
1 − 1
p
)
+
∑
χ̸=χ0
χ(a) log L (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... |
4.3 Dirichlet L -functions 127
Here each term in the product is 1 + O (1/x ), and the number of factors is
≤ ω(q ), so the product is 1 + Oq (1/x ), and hence the above is
= eC0 ϕ(q )
q (log x )
(
1 + Oq
(1
log x
))
.
T o complete the proof it suffices to combine this with Theorem 4.11(d)
in (4.27). □
4.3.1 Exercises
1.... | {
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128 Primes in arithmetic progressions: I
(d) Write ∑
d ≤x = ∑
d ≤y + ∑
y<d ≤x = S1 + S2 where 1 ≤ y ≤ x . Use
part (b) to show that S1 = 1
2 xL (1,χ ) + Oχ(x /y) + O ( y2 /x ).
(e) Use the results of part (a) to show that S2 ≪χ f (x /y).
(f) By making an appropriate choice of y, deduce that if χis a non-principal
chara... | {
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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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4.3 Dirichlet L -functions 129
(c) Show that if x ≥ C , C ≥ 2, and ( a,q ) = 1, then
∑
x /C <p≤x
p≡a (q )
log p
p = log C
ϕ(q ) + Oq (1).
(d) Show that for any positive integer q there is a small number cq and a
large number Cq such that if x ≥ 2Cq and ( a,q ) = 1, then
∑
x /Cq <p≤x
p≡a (q )
log p
p > cq .
(e) Show tha... | {
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130 Primes in arithmetic progressions: I
(e) Deduce that
d (n) = n(log 2 +o(1))/log log n
as y →∞ .
7. Let R(n) denote the number of ordered pairs a,b such that a2 + b2 = n
with a ≥ 0 and b > 0. Also, let r (n) denote the number of such pairs for
which ( a,b) = 1. Finally , let χ−4 =
(−4
n
)
be the non-principal charac... | {
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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... |
4.3 Dirichlet L -functions 131
8. Let K = Q(
√−1) be the Gaussian field, OK ={ a + ib : a,b ∈ Z} the ring
of integers in K . Ideals a in OK are principal, a = (a + ib ), and have norm
N (a) = a2 + b2 .
(a) Explain why the number of ideals a with N (a) ≤ x is π
4 x + O (x 1/2 ).
(b) For σ> 1, let ζK (s) = ∑
a N (a)−s be ... | {
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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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132 Primes in arithmetic progressions: I
(d) Deduce that /Phi1 q (z) has a coefficient whose absolute value is at least
exp
(
q (log 2 −ε)/log log q )
if y > y0 (ε).
10. Gr ¨ossencharaktere for Q(√−1), continued from Exercise 4.2.7.
(a) For σ> 1 put
L (s,χm ) =
∑
α∈OK
′
χm (α) N (α)−s = 1
4
∑
a,b∈Z
(a,b)̸=(0,0)
χm (a + ... | {
"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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4.4 Notes 133
4.4 Notes
Section 4.1. Ramanujan’s sum was introduced by Ramanujan (1918). Incredi-
bly , both Hardy and Ramanujan missed the fact thatcq (n) be written in closed
form: The formula on the extreme right of (4.7) is due to H ¨ older (1936). Nor-
mally one would say that a functionf is even if f (x ) = f (−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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134 Primes in arithmetic progressions: I
and thus to show that L (1,χp ) ̸=0 it suffices to show that Q ̸=1. Dirichlet
established this by means of Gauss’s theory of cyclotomy . Accounts of this are
found in Davenport (2000, Sections 1–3), and in Narkiewicz (2000, pp. 64–
65). An alternative proof thatQ ̸=1 was given mo... | {
"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... |
4.5 References 135
(1997). A theorem of Ingham implying that Dirichlet’s L -functions have no zeros
with real part one, Enseignement Math . (2) 43, 281–284.
Bateman, P . T ., Pomerance, C., & V aughan, R. C. (1981). On the size of the coefficients
of the cyclotomic polynomial, Coll. Math. Soc. J. Bolyai , pp. 171–202.
C... | {
"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... |
136 Primes in arithmetic progressions: I
(1895b). ¨Uber das Nichtverschwinden Dirichletscher Reihen mit reelen Gliedern,
Sitzber . Kais. Akad. Wiss. Wien 104, 2a, 1158–1166.
(1897). ¨Uber Multiplikation und Nichtverschwinden Dirichlet’scher Reihen, J. Reine
Angew . Math. 117, 169–184.
(1899). Eine asymptotische Aufgabe... | {
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5
Dirichlet series: II
5.1 The inverse Mellin transform
In Chapter 1 we saw that we can express a Dirichlet series α(s) = ∑ ∞
n=1 an n−s
in terms of the coefficient sum A(x ) = ∑
n≤x an , by means of the formula
α(s) = s
∫ ∞
1
A(x )x −s−1 dx , (5.1)
which holds for σ> max(0,σc ). This is an example of a Mellin transform... | {
"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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138 Dirichlet series: II
Theorem 5.1 (Perron’s formula) If σ0 > max(0,σc ) and x > 0, then
∑
n≤x
′
an = lim
T →∞
1
2πi
∫ σ0 +iT
σ0 −iT
α(s) x s
s ds .
Here ∑ ′ indicates that if x is an integer , then the last term is to be counted with
weight 1/2.
Proof Choose N so large that N > 2x + 2, and write
α(s) =
∑
n≤N
an n−s ... | {
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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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5.1 The inverse Mellin transform 139
W e have now established a precise relationship between (5.1) and (5.2), but
Theorem 5.1 is not sufficiently quantitative to be useful in practice. W e express
the error term more explicitly in terms of thesine integral
si(x ) =−
∫ ∞
x
sin u
u du .
By integration by parts we see 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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140 Dirichlet series: II
since the integrand has a pole with residue 1 at s = 0. In addition,
∫ σ0 ±iT
−∞±iT
ys ds
s =
∫ σ0
−∞
yσ±iT
σ ± iT d σ ≪ 1
T
∫ σ0
−∞
yσ d σ = yσ0
T log y ≪ yσ0
T ,
so we have (5.9) in the case y ≥ 2. The case y ≤ 1/2 is treated similarly , but
the contour is taken to the right, and there is no ... | {
"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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5.1 The inverse Mellin transform 141
In classical harmonic analysis, for f ∈ L1 (T) we define Fourier coefficients
ˆf (k) =
∫1
0 f (x )e(−kα) d α, and we expect that the Fourier series ∑ ˆf (k)e(kα)
provides a useful formula for f (α). As it happens, the Fourier series may
diverge, or converge to a value other than f (α)... | {
"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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142 Dirichlet series: II
Mellin transform of α(s)/s. Further instances of this pairing arise if we take a
weight function w(x ), and form a weighted summatory function
Aw(x ) =
∞∑
n=1
an w(n/x ).
Let K (s) denote the Mellin transform of w(x ),
K (s) =
∫ ∞
0
w(x )x s−1 dx .
Then we expect that
α(s)K (s) =
∫ ∞
0
Aw(x )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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5.1 The inverse Mellin transform 143
by the calculus of residues; this is
=
k∑
j =0
(−1) j y− j
j !(k − j )! = 1
k! (1 − 1/y)k
by the binomial theorem.
2. Riesz typical means.For positive integers k and positive real x put
Rk (x ) = 1
k!
∑
n≤x
an (log x /n)k . (5.20)
Then Rk (x ) =
∫x
0 Rk−1 (u)/ud u where R0 (x ) = A(... | {
"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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144 Dirichlet series: II
of analytic continuation it follows that (5.23) holds for σ> max(0,σc ). In the
opposite direction,
P (x ) = 1
2πi
∫ σ0 +i ∞
σ0 −i ∞
α(s)Ŵ(s)x −s ds (5.25)
for x > 0, σ> max(0,σc ). T o prove this we recall from Theorem 1.5 that
α(s) ≪ τ uniformly for σ ≥ ε+ max(0,σc ), and from Stirling’s form... | {
"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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5.1 The inverse Mellin transform 145
5.1.1 Exercises
1. Show that if σc <σ 0 < 0, then
lim
T →∞
1
2πi
∫ σ0 +iT
σ0 −iT
α(s) x s
s ds =
∑ ′
n>x an .
2. (a) Show that if y ≥ 0, then
− π
2 = si(0) ≤ si( y) ≤ si(π) = 0.28114 ....
(b) Show that if y ≥ 0, then
ℑ
∫ ∞
y
eiu
u du = ℑ
∫ y+i ∞
y
eiz
z dz .
(c) Deduce that if y ≥ 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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146 Dirichlet series: II
6. Let w(x ) ր, and suppose that w(x ) ≪ x σ as x →∞ for some fixed σ.
Let σw be the infimum of those σ such that
∫∞
0 w(x )x −σ−1 dx < ∞, and
put
K (s) =
∫ ∞
0
w(x )x −s−1 dx
for σ>σ w.
(a) Show that Aw(x ) = ∑ ∞
n=1 an w(x /n) satisfies Aw(x ) ≪ x θ for θ>
max(σw,σc ).
(b) Show that
K (s)α(s) =
... | {
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5.2 Summability 147
This is a form of the Heisenberg uncertainty principle. From it we see that
iff tends to 0 rapidly outside [ − A, A], and if ˆf tends to 0 rapidly outside
[−B,B ], then AB ≫ 1.
9. (a) Note the identity
f
g = 1
2 | f + g|2 − 1
2 | f − g|2 + i
2 | f + ig |2 − i
2 | f − ig |2 .
(b) Show that if f ∈ L 1... | {
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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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148 Dirichlet series: II
the above by saying that if lim N →∞ sN = a, then
lim
N →∞
1
N
N∑
n=1
sn = a. (5.28)
Here, as in Abel summability and in most other summabilities, each term in
the second limit is a linear function of the terms in the first limit. Following
T oeplitz and Schur, we characterize those linear trans... | {
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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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5.2 Summability 149
T o establish the second assertion, suppose that ε> 0 and that |an | <ε for
n > N = N (ε). Now
|bm |≤
N∑
n=1
|tmn an |+
∑
n>N
|tmn an |= /Sigma1 1 + /Sigma1 2 ,
say . From (5.29) and the argument above with A = ε we see that /Sigma1 2 ≤ C ε.
From (5.30) we see that lim m→∞ /Sigma1 1 = 0. Hence lim 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",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
150 Dirichlet series: II
The converse of Abel’s theorem on power series is false, but T auber (1897)
proved a partial converse: If an = o(1/n) and ∑ an = a (A), then ∑ an = a.
Following Hardy and Littlewood, we call a theorem ‘T auberian’ if it provides
a partial converse of an Abelian theorem. The qualifying hypothesi... | {
"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... |
5.2 Summability 151
Arguing as we did before, we find that
lim inf
N →∞
N∑
n=1
an ≥ a − Aε/(1 − ε),
so that
lim inf
N →∞
N∑
n=1
an ≥ a,
and the proof is complete. □
If we had argued from (a) or (b), then the treatment of the term T3 above
would have been simpler, since from (a) it follows that T3 ≥ 0, while from
(b) 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",
"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
152 Dirichlet series: II
converges for every δ> 0. Let β be fixed, β ≥ 0, and suppose that
I (δ) = (α + o(1))δ−β (5.35)
as δ → 0+. If, moreover , there is a constant A ≥ 0 such that
a(u) ≥− A(u + 1)β−1 (5.36)
for all u ≥ 0, then
∫ U
0
a(u) du =
( α
Ŵ(β + 1) + o(1)
)
U β. (5.37)
The basic properties of the gamma function... | {
"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... |
5.2 Summability 153
and
| P±(x ) − χJ (x )|≤ εx (1 − x ) + 5χK(x ). (5.41)
Proof Let g(x ) = (χJ (x ) − x )/(x (1 − x )). Then g is continuous in [0 ,1]
apart from a jump discontinuity at x = 1/e of height e2 /(e − 1) < 5. Hence
by W eierstrass’s theorem on the uniform approximation of continuous func-
tions by polynom... | {
"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... |
154 Dirichlet series: II
Since ε can be arbitrarily small, we deduce that
lim sup
U →∞
U −β
∫ U
0
a(u) du ≤ 0.
By arguing similarly with P− instead of P+, we see that the corresponding
liminf is ≥ 0, and so we have (5.37) in the case α = 0.
Suppose now that α ̸=0, β> 0. W e note first that
∫ ∞
0
(u + 1)β−1 e−uδ du = eδ
... | {
"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... |
5.2 Summability 155
Proof Ta k e β = 1, p(z) = ∑ ∞
n=0 sn zn = (1 − z)−1 ∑ ∞
n=0 an zn in Corollary
5.9. Then ∑ N
n=0 sn = (α + o(1)) N , which is the desired result. □
For Dirichlet series we have similarly
Theorem 5.11 Suppose that α(s) = ∑ ∞
n=1 an n−s converges for σ> 1, and
that β ≥ 0.I f α(σ) = (α + o(1))(σ − 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... |
156 Dirichlet series: II
This is best possible (take an = 1 + n−1/2 ), but if the error term is oscilla-
tory , then smoothing may reduce its size (consider an = cos √n). Conversely if
(5.45) holds and if the sequence an is bounded, then the method used to prove
Theorem 5.6 can be used to show that
N∑
n=1
an = N + 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... |
5.2 Summability 157
5.2.1 Exercises
1. Let T be a regular matrix such that tmn ≥ 0 for all m,n. Show that if
limn→∞ an =+ ∞ , then lim m→∞ bm =+ ∞ .
2. Show that if T = [tmn ] and U = [umn ] are regular matrices, then so is
TU = V = [vmn ] where
vmn =
∞∑
k=1
tmk ukn .
3. Show that if b = T a and lim m→∞ bm = a whenever... | {
"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... |
158 Dirichlet series: II
(a) Show that
m∑
n=k
tmn = m!
mk (m − k)!
for 1 ≤ k ≤ m .
(b) V erify that T is regular.
(c) Show that if an = ∑ n
k=0 x k /k! for n ≥ 0, then bm = (1 + x /m)m for
m ≥ 1.
9. (Mercer’s theorem) Suppose that
bm = 1
2 am + 1
2 · a1 + a2 +···+ am
m
for m ≥ 1. Show that
an = 2n
n + 1 bn − 2
n(n + 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... |
5.2 Summability 159
(a) Show that if
an =
n∑
k=1
ck
(
1 − k
n + 1
)
for n ≥ 1, then the bm given in (5.32) satisfies
bm =
m∑
k=1
ck
(
1 − log k
log(n + 1)
)
.
(b) Show that tmn ≥ 0 for all m,n.
(c) Show that
∞∑
n=1
tmn = 1 + log 2
log(m + 1) .
(d) Show that lim m→∞ tmn = 0.
(e) Conclude that if ∑ ck = c (C, 1), then ∑ c... | {
"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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160 Dirichlet series: II
(b) Show that if an is a bounded sequence and |z| < 1, then
∞∑
n=1
nan zn
1 − zn =
∞∑
n=1
(∑
d |n
da d
)
zn .
(c) Show that ∑ ∞
n=1 µ(n)/n = 0 (L).
(d) Deduce that if ∑ ∞
n=1 µ(n)/n converges, then its value is 0. (See (6.18)
and (8.6).)
(e) Show that ∑ ∞
n=1 (/Lambda1 (n) − 1)/n =− 2C0 (L).
(f... | {
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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... |
5.2 Summability 161
(c) Let B ( N ) = ∑ N
n=1 nan . Show that if ∑ an converges, then B ( N ) =
o( N )a s N →∞ .
(d) Show that if P (δ) converges for δ> 0, then
sN − P (1/N ) = B ( N )
N +
∫ N
1
B (u)
(1
u2 − e−u/N
u2 − e−u/N
uN
)
du
+
∫ ∞
N
B (u)e−u/N
(u
N − 1
)du
u2 .
(e) Show that if B ( N ) = o( N ), then sN − P (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... |
162 Dirichlet series: II
27. Suppose that for every ε> 0 there is an η> 0 such that∑
N <n≤(1+η) N |an | <ε whenever N > 1/η. Show that if ∑ an = a (A),
then ∑ an = a.
28. Show that if ∑ an = a (C, 1) and if an+1 − an = O (|an |/n), then ∑ an = a.
29. (Hardy & Littlewood 1913, Theorem 27) Show that if ∑ an = a (A) and i... | {
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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... |
5.3 Notes 163
in the same half-plane, if and only if the values taken by α(s) in this half-
plane are bounded away from 0. Ingham (1962) noted a fallacy in Zygmund’s
account of L´ evy’s theorem, corrected it, and gave an elementary proof of the
generalization to absolutely convergent Dirichlet series. See also Goodman ... | {
"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... |
164 Dirichlet series: II
as a special case. Wiener’s theory is discussed by Hardy (1949), Pitt (1958), and
Widder (1946). Among the longer expositions of T auberian theory , the recent
accounts of Korevaar (2002, 2004) are especially recommended.
5.4 References
Apostol, T . (1976). Modular Functions and Dirichlet Serie... | {
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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... |
5.4 References 165
(1914b). Some theorems concerning Dirichlet’s series, Messenger Math. 43, 134–147;
Collected P apers, V ol. 6. Oxford: Clarendon Press, 1974, pp. 542–555.
(1926). A further note on the converse of Abel’s theorem, Proc. London Math.
Soc. (2) 25, 219–236; Collected P apers , V ol. 6. Oxford: Clarendon ... | {
"creation_date": "2025-02-03",
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"file_path": "/home/suryaremanan/Documents/IMO/augmentoolkit/original/input/(Cambridge Stud... |
166 Dirichlet series: II
(1908). Zwei neue Herleitungen f ¨ ur die asymptotische Anzahl der Primzahlen unter
einer gegebenen Grenze, Sitzungsberichte Akad. Wiss . Berlin 746–764; Collected
W orks, V ol.4. Essen: Thales V erlag, 1986, pp. 21–39.
(1909). Handbuch der Lehre von der V erteilung der Primzahlen , Leipzig: T ... | {
"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... |
5.4 References 167
Titchmarsh, E. C. (1939). The Theory of Functions , Second Edition. Oxford: Oxford
University Press.
(1986). The Theory of the Riemann Zeta-function , Second Edition. Oxford: Oxford
University Press.
T oeplitz, O. (1911). ¨Uber algemeine lineare Mittelbildungen , W arsaw: Prace mat–fiz
22, 113–119.
Wi... | {
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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... |
6
The Prime Number Theorem
6.1 A zero-free region
The Prime Number Theorem (PNT) asserts that
π(x ) ∼ x
log x
as x tends to infinity . W e shall prove this by using Perron’s formula, but in
the course of our arguments it will be important to know that ζ(s) ̸=0 for
σ ≥ 1. In Chapter 1 we saw that ζ(s) ̸=0 for σ> 1, but i... | {
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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... |
6.1 A zero-free region 169
put
g(z) = f (z)
K∏
k=1
R2 − z zk
R(z − zk ) .
The kth factor of the product has been constructed so that it has a pole at zk , and
so that it has modulus 1 on the circle |z|= R. Hence g is an analytic function
in the disc |z|≤ R, and if |z|= R, then |g(z)|=| f (z)|≤ M . Hence by the
maximum ... | {
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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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170 The Prime Number Theorem
Moreover, if k > 0, then
∫ 1
0
h( Re (θ))e(kθ) d θ = R−k
2πi
∮
|z|=R
h(z)zk−1 dz = 0,
and
∫ 1
0
h( Re (θ))e(−kθ) d θ = Rk
2πi
∮
|z|=R
h(z)z−k−1 dz = Rk h(k) (0)
k! .
By forming a linear combination of these identities we see that if k > 0, then
∫ 1
0
h( Re (θ))(1 + cos 2 π(kθ + φ)) d θ = Rk... | {
"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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6.1 A zero-free region 171
By Lemma 6.1 we know that
K ≤ log M/| f (0)|
log 1 /R ≪ log M
| f (0)| . (6.2)
If |z|= R, then each factor in the product has modulus 1. Consequently |g(z)|≤
M when |z|= R, and by the maximum modulus principle |g(z)|≤ M for |z|≤
R. W e also note that
|g(0)|=| f (0)|
K∏
k=1
R
|zk | ≥| f (0)|.
... | {
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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... |
172 The Prime Number Theorem
If the zeta function were to have a zero of multiplicity m at 1 + i γ, then we
would have
ζ′
ζ (1 + δ + i γ) ∼ m
δ
as δ → 0+. But
ℜ ζ′
ζ (1 + δ + i γ) =−
∞∑
n=1
/Lambda1 (n)n−1−δ cos(γ log n),
and in the very worst case this could be no larger than
∞∑
n=1
/Lambda1 (n)n−1−δ =− ζ′
ζ (1 + δ) ∼... | {
"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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6.1 A zero-free region 173
Proof Since ζ(s) is given by the absolutely convergent product (1.17) for
σ> 1, it suffices to consider σ ≤ 1. From (1.24) we see that
⏐
⏐
⏐
⏐ζ(s) − s
s − 1
⏐
⏐
⏐
⏐≤| s|
∫ ∞
1
u−σ−1 du = |s|
σ (6.5)
for σ> 0. From this we see that ζ(s) ̸=0 when σ> |s − 1|, i.e., in the parabolic
region σ> (1 +... | {
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174 The Prime Number Theorem
a0 + a1 cos 2 πθ +···+ aN cos 2 πN θ, the ratio a1 /a0 can be arbitrarily close
to 2, as we see in the Fej´ er kernel
/Delta1 N (θ) = 1 + 2
N −1∑
n=1
(
1 − n
N
)
cos 2 nπθ = 1
N
(sin πN θ
sin πθ
)2
≥ 0,
but it must be strictly less than 2 since
a0 − 1
2 a1 =
∫ 1
0
T (θ)(1 − cos 2 πθ) d θ> 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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6.1 A zero-free region 175
Since |s − ρ|≍| s1 − ρ| for all zeros ρ in the sum, it follows that
1
s − ρ − 1
s1 − ρ ≪ 1
|s1 − ρ|2 log τ ≪ℜ 1
s1 − ρ.
Now (6.6) follows on combining this with (6.9) and (6.10) in (6.11).
T o derive (6.7) we begin as in our proof of (6.6). From Corollary 1.11 and
the triangle inequality we s... | {
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176 The Prime Number Theorem
6.1.1 Exercises
1. (a) Show that if |z| < R, |w|≤ R, and z ̸=w, then
⏐
⏐
⏐
⏐
z
w− R2
(z − w) R
⏐
⏐
⏐
⏐≥ 1.
(b) Show that if |w|≤ ρ< R, |z|= r < R, and z ̸=w, then
⏐
⏐
⏐
⏐
z
w− R2
(z − w) R
⏐
⏐
⏐
⏐≥ rρ + R2
(r + ρ) R .
(c) Suppose that f is analytic in the disc |z|≤ R.F o r r ≤ R put M (r ) ... | {
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6.1 A zero-free region 177
5. Let χ0 denote the principal character (mod 4), and χ1 the non-principal
character (mod 4).
(a) Show thatL (1,χ1 ) = π/4, and hence that there is a neighbourhood of
1 in which L (s,χ1 ) ̸=0.
(b) Show that if σ> 1, then
ℜ
(
−3 L ′
L (σ,χ0 ) − 4 L ′
L (σ + it ,χ1 ) − L ′
L (σ + 2it ,χ0 )
)
≥ ... | {
"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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178 The Prime Number Theorem
(d) Show that if 1 <σ ≤ 2, then
⏐
⏐
⏐
⏐
ζ(σ + i (t + 1))
ζ(σ + it )
⏐
⏐
⏐
⏐≤
∞∏
k=−∞
|ζ(σ + ik )|2 ˆf (k)
uniformly for all real t .
(e) Show that if σ> 1, then
(σ − 1)4/π ≪
⏐
⏐
⏐
⏐
ζ(σ + i (t + 1))
ζ(σ + it )
⏐
⏐
⏐
⏐≪ (σ − 1)−4/π
uniformly in t .
(f) Show that
(log t )−4/π ≪
⏐
⏐
⏐
⏐
ζ(1 + ... | {
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6.2 The Prime Number Theorem 179
(d) By mimicking the proof of Theorem 6.7, but with s1 = 1 +
θ(2t + 1)/φ(2t + 1) + it , show that
ζ′
ζ (s) ≪ φ(2t + 2)
θ(2t + 2) ,
| log ζ(s)|≤ log φ(2t + 2)
θ(2t + 2) + O (1),
1
ζ(s) ≪ φ(2t + 2)
θ(2t + 2)
for σ ≥ 1 − 1
2 cθ(2t + 2)/φ(2t + 2).
10. Suppose that ζ(s) ̸=0 for σ ≥ η(t ), t ... | {
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180 The Prime Number Theorem
Here li( x )i st h e logarithmic integral,
li(x ) =
∫ x
2
1
log u du .
By integrating this integral by parts K times we see that
li(x ) = x
K −1∑
k=1
(k − 1)!
(log x )k + OK
( x
(log x )K
)
. (6.15)
On combining this with (6.14) we see that
π(x ) = x
log x + O
( x
(log x )2
)
.
This is a qu... | {
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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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6.2 The Prime Number Theorem 181
Put σ1 = 1 − c/log T where c is a small positive constant, and let C denote
the closed contour that consists of line segments joining the points σ0 − iT ,
σ0 + iT , σ1 + iT , σ1 − iT . From Theorem 6.6 we know that ζ′
ζ (s) has a simple
pole with residue −1a t s = 1, but that it is othe... | {
"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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