The theory of numbers, by Robert D. Carmichael ...

CHAPTER II

ON THE INDICATOR OF AN INTEGER
~ 14. DEFINITION. INDICATOR OF A PRIME POWER
DEFINITION. If m is any given positive integer the number of positive integers not greater than mn and prime to it is
called the indicator of m. It is usually denoted by p(m), and
is sometimes called Euler's ~-function of m. More rarely,
it has been given the name of totient of m.
As examples we have
~(I)=I, 0(2)=I, 0(3) = 2, 0(4) =2
If p is a prime number it is obvious that
(p) =p-I;
for each of the integers I, 2, 3,..., p-I is prime to p.
Instead of taking nm=p let us assume that enp", where
a is a positive integer, and seek the value of 0(pa). Obviously,
every number of the set i, 2, 3,., pa either is divisible
by p or is prime to pa. The number of integers in the set
divisible by p is p —l' Hence pa-pa-l of them are prime
to p. Hence ~(pa) =pa-pa-1. Therefore
If p is any prime number and a is any positive integer, then
O(P') == P (I I_)
~ 15. THE INDICATOR OF A PRODUCT
I. If,u and v are any two relatively prime positive integers
then
) (T) = (P G() ( (Pv4)
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