The Euler totient function, also known as Euler's phi function, is a mathematical function that counts the number of positive integers less than or equal to a given number that are relatively prime to that number. In other words, it calculates the number of numbers that are coprime to a given number.
The Euler totient function can be calculated using the formula φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk), where n is the given number and p1, p2, ..., pk are the distinct prime factors of n.
The Euler totient function has many applications in number theory, cryptography, and other branches of mathematics. It is used to solve problems related to modular arithmetic, to determine the order of elements in a group, and to calculate the totient sum, among other things.
The Euler totient function is closely related to the prime factorization of a number. This is because the number of positive integers less than or equal to a given number and relatively prime to it is equal to the product of (1 - 1/p), where p is a distinct prime factor of the given number.
No, the Euler totient function is only defined for positive integers. It is undefined for non-positive integers, as there are no positive integers less than or equal to 0 or negative numbers. However, it can be extended to complex numbers using the Dirichlet eta function.