📂Number Theory

Euler's Totient Function in Analytic Number Theory

Definition ¹

The arithmetic function defined as follows $\varphi$ is called the totient function. $$\varphi (n) := \sum_{\gcd ( k , n ) = 1} 1$$

Basic Properties

• [1] Totient series: the norm $N$. That is, $$\sum_{d \mid n } \varphi (d) = N(n)$$
• [2] Multiplicativity: For all $m, n \in \mathbb{N}$ that satisfy $\gcd (m,n) = 1$, $\varphi (mn) = \varphi (m) \varphi (n)$

Explanation

$$\begin{matrix} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \varphi(n) & 1 & 1 & 2 & 2 & 4 & 2 & 6 & 4 & 6 & 4 \\ \sum_{d \mid n} \varphi(d) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \end{matrix}$$ Yes, it is that totient function from elementary number theory. Given its numerous mysterious properties, it’s inevitably mentioned in analytic number theory as well.

Definition

[1]

Deduce directly according to the definition.

■

[2]

Deduce directly by dividing cases.

■

See Also

• The totient function in elementary number theory