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 $\phi (n)$, 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:

$\phi (n)=\prod _{i=1}^r{p}_i^{k_i}(1-\frac{1}{p_i})$.

In other words:

$\frac{\phi (n)}{n}=\prod _{i=1}^r(1-\frac{1}{p_i})$.

Behavior

Upper bound

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

Thus:

$limsup\frac{\varphi (p)}{p}=1$.

Lower bound

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

$\phi (n)\ge n^{1-ϵ}\mathrm{\forall }n\ge N_{ϵ}$.

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

Failed to parse (unknown function "\logn"): {\displaystyle \varphi(n) \ge n^{1 - (C\log\log n/\logn)}} .