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Concept# Dirichlet convolution

Summary

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
If are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution f ∗ g is a new arithmetic function defined by:
where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.
This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients:
The set of arithmetic functions forms a commutative ring, the , under pointwise addition, where f + g is defined by (f + g)(n) = f(n) + g(n), and Dirichlet convolution. The multiplicative identity is the unit function ε defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1. The units (invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0.
Specifically, Dirichlet convolution is associative,
distributive over addition
commutative,
and has an identity element,
= .
Furthermore, for each having , there exists an arithmetic function with , called the of .
The Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring. Beware however that the sum of two multiplicative functions is not multiplicative (since ), so the subset of multiplicative functions is not a subring of the Dirichlet ring. The article on multiplicative functions lists several convolution relations among important multiplicative functions.
Another operation on arithmetic functions is pointwise multiplication: fg is defined by (fg)(n) = f(n) g(n). Given a completely multiplicative function , pointwise multiplication by distributes over Dirichlet convolution: .

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