How do you prove a function is multiplicative?

An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.

Are Sigma functions multiplicative?

The function σ(x) is a multiplicative function, so its value can be determined from its value at the prime powers: Theorem. If p is prime and n is any positive integer, then σ(pn) is (pn+1-1)/(p-1).

Is the Mobius function completely multiplicative?

The Möbius function is an example of a special class of functions, called multiplicative functions. It is • completely multiplicative if f(mn) = f(m)f(n) for any m and n. • multiplicative if f(mn) = f(m)f(n) for relatively prime m and n.

What is the divisors of 12?

4. So, the divisors or factors of the number 12 are 1,2,3,4,6 and 12.

What does Multiplicatively mean?

1 : tending or having the power to multiply. 2 : of, relating to, or associated with a mathematical operation of multiplication the multiplicative property of 0 requires that a × 0 = 0 and 0 × a = 0.

Is Euler’s Phi function multiplicative?

Theorem. Euler’s phi function ϕ is multiplicative. In other words, if gcd(m, n)=1 then ϕ(mn) = ϕ(m)ϕ(n).

How do you prove that a Mobius function is multiplicative?

The Mobius function μ(n) is multiplicative. Let m and n be two relatively prime integers. We have to prove that μ(mn)=μ(m)μ(n). If m=n=1, then the equality holds.

What is Möbius function used for?

The Möbius function μ ( n ) μ(n) μ(n) is a multiplicative function which is important in the study of Dirichlet convolution. It is an important multiplicative function in number theory and combinatorics.

How do you find divisors?

Let us understand the formula of divisor when the remainder is 0, and when it is a non-zero number.