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Math 571 Analytic Number Theory I, Spring 2011, Problems 7
Due Tuesday 1st March
1. (Vaughan 1973). Suppose that for each prime
p
,0
≤
b
(
p
)
<p
and deFne
L
(
Q
)=
°
q
≤
Q
μ
(
q
)
2
±
p

q
b
(
p
)
p
−
b
(
p
)
.
(a) Prove that
L
(
Q
)=
°
m
°
r,
Ω(
r
)=
m
s
(
r
)
≤
Q
±
p
h
°
r
²
b
(
p
)
p
³
h
where
s
(
r
) is the squarefree kernel of
r
and Ω(
m
) is the total number
of prime factors of
m
.
(b) Prove that
°
r,
Ω(
r
)=
m
s
(
r
)
≤
Q
±
p
h
°
r
²
b
(
p
)
p
³
h
≥
1
m
!
°
p
≤
Q
1
/m
b
(
p
)
p
m
.
(c) Suppose that
´
p
≤
X
b
(
p
)
p
≥
F
(
X
)
>
0 for every
X>
0. Prove that
L
(
Q
)
≥
max
m
exp
µ
m
log
¶
m
−
1
F
(
Q
1
/m
)
·
¸
.
2. (Known as “Rankin’s Trick” but Rankin said that it was shown to him by Ingham!). Suppose that
Q
≥
1, and
P
∈
N
. Let
L
(
Q,P
)=
°
q
≤
Q,q

P
μ
(
q
)
2
±
p

q
b
(
p
)
p
−
b
(
p
)
.
(a) Prove that, with the obvious abuse of notation,
L
(
∞
,P
)=
¹
p

P
p
p
−
b
(
p
)
.
(b) Suppose that
θ
≥
0. Prove that
L
(
∞
,P
)
−
L
(
Q,P
)
≤
Q
−
θ
°
q

P
q
θ
μ
(
q
)
2
±
p

q
b
(
p
)
p
−
b
(
p
)
and
L
(
Q,P
)
L
(
∞
,P
)
≥
1
−
Q
−
θ
±
p

P
²
1+(
p
θ
−
1)
b
(
p
)
p
³
.
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This note was uploaded on 01/23/2012 for the course MATH 571 taught by Professor Vaughan,r during the Spring '08 term at Penn State.
 Spring '08
 Vaughan,R
 Number Theory

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