Dirichlet convolution

Dirichlet convolution

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

See also:

Johann Peter Gustav Lejeune Dirichlet

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

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(f*g)(n) = \sum_{d|n} f(d)g(n/d) </math> where the sum extends over all positive divisors d of n.

Some general properties of this operation include:

If both f and g are multiplicative, then so is f * g. (Note however that the convolution of two completely multiplicative functions need not be completely multiplicative.)
f * g = g * f (commutativity)
(f * g) * h = f * (g * h) (associativity)
f * (g + h) = f * g + f * h (distributivity)
f * ε = ε * f = f, where ε is the function defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1.
To every multiplicative f there exists a multiplicative g such that f * g = ε.

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

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L(f,s) = \sum_{n=1}^\infty \frac{f(n)}{n^s} </math> 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:

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L(f,s) L(g,s) = L(f*g,s) </math> 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.