Leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it: he … See more

The first 100 values (sequence A000010 in the OEIS) are shown in the table and graph below:
φ(n) for 1 ≤ n ≤ 100 + 1 2 3 4 5 6 7 8 9 10 0 1 1 2 2 4 2 6 4 6 4 … See more

• Note the special cases
• Compare this to the formula (See least common multiple.)
• φ(n) is even for n ≥ 3. Moreover, if n has r distinct odd prime … See more

In the words of Hardy & Wright, the order of φ(n) is "always 'nearly n'."
First
but as n goes to infinity, for all δ > 0
These two formulae … See more

There are several formulae for computing φ(n).
Euler's product formula
It states
where the product is over the distinct prime numbers dividing n. (For notation, see Arithmetical function See more

This states that if a and n are relatively prime then
The special case where n is prime is known as Fermat's little theorem.
This follows from Lagrange's theorem and the fact that φ(n) is the order of the multiplicative group of integers modulo n See more

The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as:
where the left-hand side converges for .
The See more