MATH 319 In Class Exercise on Arithmetic Functions

Names____________________________________________________

1. Recall that if f(n) is a multiplicative function, then is also multiplicative.

a) Is f(n) = n + 1 multiplicative? ______, If it is, compute F(12) = ___________.

b) Is f(n) = n² multiplicative? ________, If it is, compute F(12) =____________.

2. We showed earlier in the course that (n), the Euler function is multiplicative. Thus is multiplicative. The goal of this problem is to compute a formula for F(n).

We start by finding a formula for (n).

Since (n) is multiplicative we need a formula for (p^k) for prime p and k, a positive integer.

(2) = _____, (4) = ______, (8) = ______, (16) = ________

(3) = ______, (9) = ______, (27) = ______, (81) = ________

Using the above what should (p^k) be in terms of p and k? ____________

Consider the set {1,2,3,…p^k}. How many of these numbers are divisible by p? _____

(p^k) is the number of numbers in that set that are not divisible by p.

Thus (p^k) = _________.

Now compute F(16) = ___________.

Compute F(p^k) for prime p. ____________

What is the formula for F(n)? ___________

3. The sigma function is defined as the sum of all the positive divisors of n. Thus we have (6) = 1 + 2 + 3 + 6 = 12.

By the result mentioned in the first problem we can see that is multiplicative since f(n) = n is multiplicative. Thus to arrive at a formula for (n), we need a formula for (p^k) for prime p.

(2) = ______, (4) = ______, (8) = ______, (16) = ______

(3) = ______, (9) = ______, (27) = ______, (81) = ______

(Hint: Geometric Series)

(p^k) = __________

Compute (5040) = ___________