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PROBLEM SHEET 4
UCL TERM 1 AUTUMN 2023
(1) (a) Show that
p
Xé x
1
p
= loglogx +c +O
μ
1
logx

(x ê 3/2)
for some constant c.
(b) Show that
Z
1
x ψ(u)
u
2
du = logx +O(1) (x ê 3/2)
where ψ(x) =
(c) Show that
P
né x Λ(n).
p
Yé x
1
1 p
1
= e
b
logx +O(1) (x ê 3/2)
for some constant b.
(d) Bonus question not for submission or credit. Show that b = γ = 1
R 1
∞{t}/t
2 d t is
Euler’s constant.
(2) Show that if f is an arithmetic function that is invertible for the Dirichlet convolution
and is of polynomial growth then its inverse f
( 1) is also of polynomial growth.
[Hint: Since f
( 1) is given by a recursive formula, try a proof by induction. We
know that |f (n)| é CnA
for some A and for all n ∈ N. Try to show that f
( 1)(n) =
O(n
A+σ) (n ∈ N), where σ is a sufficiently large real number so that P dê 2 d
1
σ é
|f
C
(1)|
.
(If you use this hint, then you must prove that such σ exists.)]
E-mail address: i.petrow@ucl.ac.uk
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