Euler's totient function, often denoted as φ(n) (phi of n), is a mathematical function in number theory introduced by the Swiss mathematician Leonhard Euler. It is also known as Euler's phi function. For a positive integer "n," φ(n) represents the count of positive integers less than or equal to "n" that are coprime to "n" (i.e., they share no common positive integer divisors other than 1).

In other words, φ(n) gives the number of positive integers "k" such that 1 ≤ k ≤ n and the greatest common divisor (GCD) of "k" and "n" is 1.

The formula for Euler's totient function φ(n) is as follows:

φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pm)

where "n" is the given positive integer, and p1, p2, ..., pm are the distinct prime factors of "n."

Some properties of Euler's totient function include:

1. For a prime number "p," φ(p) = p - 1 because all numbers from 1 to p-1 are coprime to "p."

2. If "a" and "b" are coprime positive integers, then φ(a * b) = φ(a) * φ(b). This property holds due to the multiplicative property of the totient function.

3. For any prime number "p" and positive integer "k," φ(p^k) = p^k - p^(k-1), which means the number of positive integers coprime to "p^k" is equal to "p^k" minus the number of positive integers coprime to "p^(k-1)."

Euler's totient function finds applications in various areas of number theory, cryptography, and algorithms, such as the RSA encryption algorithm, which relies on properties related to the totient function.