Properties
The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition (i.e. f + g is defined by (f + g)(n)= f(n) + g(n)) and Dirichlet convolution. The multiplicative identity is the function defined by (n) = 1 if n = 1 and (n) = 0 if n > 1. The units (i.e. invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0.
Specifically, Dirichlet convolution is associative,
- (f * g) * h = f * (g * h),
distributes over addition
- f * (g + h) = f * g + f * h = (g + h) * f,
is commutative,
- f * g = g * f,
and has an identity element,
- f * = * f = f.
Furthermore, for each f for which f(1) ≠ 0 there exists a g such that f * g =, called the Dirichlet inverse of f.
The Dirichlet convolution of two multiplicative functions is again multiplicative, and every multiplicative function has a Dirichlet inverse that is also multiplicative. The article on multiplicative functions lists several convolution relations among important multiplicative functions.
Given a completely multiplicative function f then f (g*h) = (f g)*(f h), where juxtaposition represents pointwise multiplication. The convolution of two completely multiplicative functions is a fortiori multiplicative, but not necessarily completely multiplicative.
Read more about this topic: Dirichlet Convolution
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