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Dirichlet convolution

The Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory.

See also:

If f and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function f * g, the Dirichlet convolution of f and g, by
where the sum extends over all positive divisors d of n.

Some general properties of this operation include:

With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity ε, the Dirichlet ring. The units of this ring are the arithmetical functions f with f(1) ≠ 0.

Furthermore, the multiplicative functions with convolution form an abelian group with identity element ε. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

If f is an arithmetic function, one defines its L-series by

for those complex arguments s for which the series converges (if there are any). The multiplication of L-series is compatible with Dirichlet convolution in the following sense:
for all s for which the left hand side exists. This is akin to the convolution theorem if one thinks of L-series as a Fourier transform.