2301609 Analytic Number Theory I Exercise Set I
I – 1. Let li(x) = Z x
2
dt
logt. Apply l’Hˆopital’s rule to show that
xlim→∞
li(x)
x/logx = 1, and deduce that each of lim
x→∞
π(x)
li(x) = 1 and lim
x→∞
π(x)
x/logx = 1 implies each other.
Here, π(x) denotes the number of primes less than or equal tox.
I – 2. Determine the arithmetic functionf such that for every positive integern, we have µ(n) =X
d|n
f(d), i.e., is it multiplicative, and what are its values on prime powers?
I – 3. Suppose that f : N → Z is totally multiplicative, i.e., f(mn) = f(m)f(n) for all m, n∈N, with f(n) = 0 or ±1. Prove that
X
d|n
f(d)≥0 and X
d|n2
f(d)≥1.
I – 4. A positive integer n is called abundant [perfect] if σ(n) ≥ 2n [σ(n) = 2n]. Prove that any integer of the form2r−1(2r−1)is abundant, and prove also that anevennumber is perfect if and only if it is of this form with 2r−1 is prime. Show, in addition, that there is no square free perfect number apart from 6.
I – 5. Verify the relations: ζ(s)ζ(s−1)↔σ(n) and ζ(s)
ζ(2s) ↔ |µ(n)|.
I – 6. For each n∈N, let
f(n) =
n
X
m=1
n gcd(m, n). (i) Show that f(n) =X
d|n
dφ(d).
(ii) Let n=pe11pe22. . . perr >1 be the prime factorization ofn. Prove that
f(n) =
p2e1 1+1+ 1 p1+ 1
p2e2 2+1+ 1 p2+ 1
. . .
p2rer+1+ 1 pr+ 1
.
I – 7. Prove that if n is a positive integer and α is a non-negative real number, then
n−1
X
k=0
α+ k
n
= [nα].
