Lecture 3

Question 1

Define Arithmetical functions

Answer 1

A real or complex valued function defined on the positive integers is called an arithmetical function (or number theoretic function)

• f ∶ N → C.

Question 2

What is the Mobius function

Answer 2

For n > 1, write n = p^a1_1 . . . p^ak_k. Then, the Mobius function is defined as follows:

µ(n) = 1 if n = 1,
µ(n) = (−1)^k if a1 = . . . = ak = 1,
µ(n) = 0 if some ai ≥ 2.

Piecewise Answer, see equation sheet if unclear

Question 3

How can we write the sum of Mobius function over divisors

Answer 3

Sum_(d|n) µ(d) = [1/n] = {1 if n=1: 0 if n>1}

Question 4

What is the Euler Totient function

Answer 4

If n ≥ 1, the Euler-Totient function, φ(n), is
defined as
* φ(n) ∶= #{k ∶ k ≤ n and gcd(k, n) = 1}

Question 5

What is the alternative form of the Euler Totient function

Answer 5

Shows the function as a sum over the constant 1. See equation sheet

Question 6

When does ∑_(d∣n) φ(d) = n.

Answer 6

When n >= 1

Question 7

How can we equate the Mobius function and the Euler Totient function

Answer 7

φ(d) = ∑_(d∣n) µ(d) n/d

Question 8

What is the product formula for the Euler Totient function

Answer 8

See equation sheet

Question 9

Give the 5 properties of the Euler Totient function

Answer 9

a) φ(p^a) = p^a − p^(a−1) for prime p and a ≥ 1.
b) φ(mn) = φ(m)φ(n)d/φ(d), where d = gcd(m, n).
c) φ(mn) = φ(m)φ(n) if gcd(m, n) = 1.
d) a∣b Ô⇒ φ(a)∣φ(b).
e) φ(n) is even for n ≥ 3. Moreover if n has r distinct odd prime factors, then 2^r∣φ(n).