In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory. This was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician.

Definition

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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

$(f*g)(n)\; =\; \backslash sum\_\{d\backslash ,\backslash mid\backslash ,n\}\; f(d)g(n/d)\; \backslash ,$

where the sum extends over all positive divisors d of n.

Properties

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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.
• For each f for which f(1) ≠ 0 there exists a g such that f * g = ε. g is called the Dirichlet inverse of f.
• In particular, every multiplicative f has a Dirichlet inverse g that is also multiplicative.
• If f is completely multiplicative then f (g*h) = (f g)*(f h).

With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity ε, the Dirichlet ring (note that it is not a field because some arithmetic functions do not have Dirichlet inverses). 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.

Dirichlet inverse

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Given an arithmetic function f, an explicit recursive formula for the Dirichlet inverse may be given as follows:

$f^\{-1\}(1)\; =\; \backslash frac\; \{1\}\{f(1)\}$

and for $n>1$,

f\left(\frac{n}{d}\right) f^{-1}(d)

When $f(n)=1$ for all n, then the inverse is $f^\{-1\}(n)=\backslash mu(n)$, the Möbius function. More general inversion relationships are given by the Möbius inversion formula.

Dirichlet series

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If f is an arithmetic function, one defines its Dirichlet series generating function by

$$

DG(f;s) = \sum_{n=1}^\infty \frac{f(n)}{n^s}

for those complex arguments s for which the series converges (if there are any). The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense:

$$

DG(f;s) DG(g;s) = DG(f*g;s)\,

for all s for which the left hand side exists. This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.

References

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• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0387901639

Arithmetic functions | Binary operations

Zahlentheoretische Funktion | 디리클레 합성곱 | Dirichletfaltning