Multiplicative function

In number theory, also known as arithmetic, an arithmetic function, denoted f(m), (that is, one defined for m integer) is called multiplicative if f(1) = 1 and also satisfies that f(m n) = f(m) f(n) when m and n are coprime integers (they have no common factors). In a multiplicative function the image of the product is equal to the product of the images. In this way, a multiplicative function is determined whenever the value it assumes for the powers of the prime numbers is known.

Function φ φ {displaystyle phi } of Euler is called multiplier functionsince (c,d)=1⇒ ⇒ φ φ (cd)=φ φ (c)⋅ ⋅ φ φ (d){displaystyle (c,d)=1Rightarrow phi (cd)=phi (c)cdot phi (d)}. There are several functions in the theory of numbers that possess this property.

Among the multiplicative functions are the fully multiplicative functions, which are those that also satisfy f(m n) = f(m) f(n) when m and n are not coprime to each other.

Using multiplicative functions as Dirichlet series development coefficients, complex functions are obtained, the study of which provides relevant information about the distribution of numbers. An example of this is the relationship of the most classical arithmetic functions with the Riemann zeta function.

Examples

Some examples of multiplicative functions that are relevant in number theory are:

• φ(n): the φ function of Euler, which counts the positive integers coprimos with n.
• μ(n): the function of Möbius, related to the number of prime factors of the numbers not divisible by a perfect square.
• d(n): the number of positive dividers n.
• σ(n): the sum of all positive dividers n.
• The function that calculates sum of all the powers of order k of positive dividers n (the σ function is the case with k=1 and function d the case with k=0).
• If we represent by r₂(n) to the added function of squares, which counts the number of different pairs of integers (a,b) such that n=a2+b2, then function r₂(n)/4 is a multipliative function.
• The function that is obtained as a product of Dirichlet is multiplied.
• The rule of a complex. N(z×u) = N(z)×N(u) where z and u are complex numbers, N(Z) = z×z*, being z* the conjugate of z.

Propositions

First

Yeah. f(n){displaystyle f(n)}is a multipliative function and

F(n)=␡ ␡ d日本語nf(d),{displaystyle F(n)=sum _{d presumption}f(d),}

then. F(n){displaystyle F(n)} is a multipliative function.

Second

␡ ␡ d日本語nφ φ (d)=n{displaystyle sum _{dsharing}phi (d)=n}

Where φ φ (k){displaystyle phi (k)} is the function φ φ {displaystyle phi } from Euler.