We encounter two interesting number-theoretic functions, 
and 
,
and discover an interesting relationship between these and the prime
factorization of a number.
The concept of a multiplicative function is also introduced, which will prove useful (now and later on).
Number-theoretic function: any function whose domain is the set of
natural numbers (and whose range is generally also in 
).
Definition 6.1: Given a positive integer n, let 
denote the
number of positive divisors of n, and 
denote the sum of those
divisors.
The notation
means ``sum the values of f as d runs over the divisors of n''. Given that, then
and
Example: : Evaluate 
and 
.
Example: : Evaluate 
and 
.
Example: : What are 
and 
when p is prime?
Example: #15, p. 110
Example: : How do 
and compare?
Definition 6.2: A number-theoretic function is said to be multiplicative if
whenever 
.
Examples: f(n)=1, f(n)=n.
By induction,
whenever the 
are pairwise relatively prime. Hence, a multiplicative
function is completely determined for n once its values on the prime powers
of the factorization of n are known:
Example: #17, p. 110
Theorem 6.1
If 
is the prime factorization of n>1,
then the positive divisors of n are precisely those integers of the form
, where 
for i in
.
Theorem 6.2
If is the prime factorization of n>1,
then
-
and
-
The notation
means ``multiply the values of f as i runs over from 1 to r''. Given that, then
and
Let's check for n=240.
Theorem 6.3 The functions and are multiplicative
functions.
Lemma If , then the set of positive divisors of mn
consists of all products 
, where 
, 
, and
; furthermore these products are all distinct.
Theorem 6.4 If f is a multiplicative function and F is defined by
then F is also multiplicative.
Corollary: the functions and are multiplicative
functions.