HOMEWORK SHEET 8 - UIUC MATH 453
NICOLAS ROBLES
Exercise 0.1. Let n ∈ N. Prove that
n
σ(n!) X 1
≥
.
n!
i
i=1
Exercise 0.2. Let n1 , n2 , · · · , nm be distinct even perfect numbers. Prove that
φ(n1 n2 · · · nm ) = 2m−1 φ(n1 )φ(n2 ) · · · φ(nm ).
Exercise 0.3. We begin with the following definition.
Definition 0.1. Let n ∈ N. Then n is said to be almost perfect if σ(n) = 2n − 1.
Let k ∈ N. Prove that any integer of the form 2k is almost perfect. (Note: the only known
almost perfect numbers are of this form.)
Exercise 0.4. Prove that if n is an odd perfect number, then n = pa m where p, a and m are
positive integers with p an odd prime number not dividing m, p ≡ a ≡ 1 mod 4, and m a perfect
square. Conclude that any odd perfect number is congruent to 1 modulo 4.
Exercise 0.5. Let n ∈ N and let ω(n) denote the number of distinct prime numbers dividing n.
Prove that
X
|µ(n)| = 2ω(n) .
d|n
Exercise 0.6. In class we proved the theorem that if f is an arithmetic function and if we define
the divisor sum
X
F (n) =
f (d),
d|N
then one has that: if f is multiplicative, then so is F . The proof was left unfinished. The following
exercise closes the gap. Let m and n be positive integers with (m, n) = 1. Prove that each divisor
d > 0 of mn can be written uniquely as d1 d2 where d1 , d2 > 0, d1 |m, d2 |n, and (d1 , d2 ) = 1 and for
each such product d1 d2 corresponds to a divisor d of mn.
1