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).
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Definition

Let $n$ be a natural number. The Euler phi-function or Euler totient function of $n$, denoted $\varphi (n)$, 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:

$\varphi (n)=\prod _{i=1}^{r}p_{i}^{k_{i}}\left(1-{\frac {1}{p_{i}}}\right)$.

In other words:

${\frac {\varphi (n)}{n}}=\prod _{i=1}^{r}\left(1-{\frac {1}{p_{i}}}\right)$.

Behavior

Upper bound

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

Thus:

$\lim \sup {\frac {\phi (p)}{p}}=1$.

Lower bound

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

$\varphi (n)\geq n^{1-\epsilon }\ \forall \ n\geq N_{\epsilon }$.

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

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

Relation with other arithmetic functions

• Universal exponent (also called Carmichael function) is the exponent of the multiplicative group modulo $n$. The universal exponent of $n$, usually denoted $\lambda (n)$, divides $\varphi (n)$.

Relations expressed in terms of Dirichlet products

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

$\sum _{d|n}\varphi (d)=n$.

• $\varphi =E*\mu$: 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:

$\sum _{d|n}d\mu (n/d)=\varphi (n)$.

• $\varphi *\tau =\sigma$: In other words, the Dirichlet product of the Euler phi-function and the divisor count function equals the divisor sum function:

$\sum _{d|n}\varphi (d)\tau (n/d)=\sigma (n)$.

• $\sigma *\varphi =E*E$.

Relation with properties of numbers

• Prime number: A number $n$ such that $\varphi (n)=n-1$.

Properties

The Euler phi-function is a multiplicative function but not a completely multiplicative function.