Category: Multiplicative functions

Lehmer's totient problem
In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem. It is known that φ(n) = n − 1
Greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greates
Completely multiplicative function
In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also
Unit function
In number theory, the unit function is a completely multiplicative function on the positive integers defined as: It is called the unit function because it is the identity element for Dirichlet convolu
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever a and b are coprime. An arithmetic function f(n) is said
Dedekind psi function
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes dividing (By convention, , which is the empty
Ramanujan tau function
The Ramanujan tau function, studied by Ramanujan, is the function defined by the following identity: where q = exp(2πiz) with Im z > 0, is the Euler function, η is the Dedekind eta function, and the f
Liouville function
The Liouville Lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers, and −1 if
Jordan's totient function
Let be a positive integer. In number theory, the Jordan's totient function of a positive integer equals the number of -tuples of positive integers that are less than or equal to and that together with
Möbius function
The Möbius function μ(n) is an important multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous
Radical of an integer
In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: The ra
Carmichael's totient function conjecture
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states