Euler totient function

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

Let $n$ be a natural number. The Euler phi-function or Euler totient function of $n$ , denoted $φ (n)$ , is defined as following:

It is the order of the multiplicative group modulo $n$ , i.e., the multiplicative group of the ring of integers modulo $n$ .
It is the number of elements in ${1, 2, \dots, n}$ that are relatively prime to $n$ .

In terms of prime factorization

Suppose we have the following prime factorization of $n$ :

$n = p_{1}^{k_{1}} p_{2}^{k_{2}} \dots p_{r}^{k_{r}}$ .

Then, we have:

$φ (n) = \prod_{i = 1}^{r} p_{i}^{k_{i}} (1 - \frac{1}{p_{i}})$ .

In other words:

$\frac{φ (n)}{n} = \prod_{i = 1}^{r} (1 - \frac{1}{p_{i}})$ .

Behavior

Upper bound

The largest values of $ϕ (n)$ are taken when $n$ is prime. $ϕ (p) = p - 1$ for a prime $p$ .

Thus:

$lim {sup}_{n \to \infty} \frac{ϕ (n)}{n} = 1$ .

Lower bound

For any $ϵ > 0$ , there exists $N_{ϵ}$ such that:

$φ (n) \geq n^{1 - ϵ} \forall n \geq N_{ϵ}$ .

With the prime number theorem, we can find a constant $C$ such that:

$φ (n) \geq n^{1 - (C \log \log n / \log n)}$ .

Summatory function and average value

Summatory function

The summatory function of the Euler phi-function is termed the totient summatory function.

Relation with other arithmetic functions

Similar functions

Universal exponent (also called Carmichael function) is the exponent of the multiplicative group modulo $n$ . The universal exponent of $n$ , usually denoted $λ (n)$ , divides $φ (n)$ .
Dedekind psi-function is similar tothe Euler phi-function, and is defined as:

$ψ (n) = n \prod_{p | n} (1 + \frac{1}{p})$ .

Relations expressed in terms of Dirichlet products

$φ * U = E$ : In other words, the Dirichlet product of the Euler phi-function and the all ones function is the identity function:

$\sum_{d | n} φ (d) = n$ .

$φ = E * μ$ : This is obtained by applying the Mobius inversion formula to the previous identity. The Euler phi-function is thus the Dirichlet product of the identity function and the Mobius function:

$\sum_{d | n} d μ (n / d) = φ (n)$ .

$φ * σ_{0} = σ$ : In other words, the Dirichlet product of the Euler phi-function and the divisor count function equals the divisor sum function:

$\sum_{d | n} φ (d) σ_{0} (n / d) = σ (n)$ .

$σ * φ = E * E$ .

Relation with properties of numbers

Prime number: A natural number $n$ such that $φ (n) = n - 1$ .
Polygonal number: A natural number $n$ such that $φ (n)$ is a power of $2$ , or equivalently, such that the regular $n$ -gon is constructible using straightedge and compass.

Properties

Multiplicativity

This arithmetic function is a multiplicative function: the product of this function for two natural numbers that are relatively prime is the value of the function at the product.
View a complete list of multiplicative functions

Complete multiplicativity

NO: This arithmetic function is not a completely multiplicative function: in other words, the product of the values of the function at two natural numbers need not equal the value at the product.

Preservation of divisibility

This arithmetic function is a divisibility-preserving function: if one natural number divides another, the value of the function at the first number also divides the value at the second number.
View other divisibility-preserving functions

If $m$ divides $n$ , then $φ (m)$ divides $φ (n)$ .

Algebraic significance

The Euler phi-function of $n$ is important in the following ways:

It is the number of generators of the cyclic group of order $n$ .
It is the order of the multiplicative group of the ring of integers modulo $n$ (in fact, this multiplicative group is precisely the set of generators of the additive group).