Dirichlet convolution

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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.

Contents 1 Definition 2 Properties 3 Weak multiplicative functions 3.1 Proof 4 Dirichlet inverse 5 Dirichlet series 6 References

[edit] Definition

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.

[edit] Properties

Some general properties of this operation include:

• (f * g) * h = f * (g * h) (associativity)
• f * δ = δ * f = f, where δ is the function defined by δ(n) = 1 if n = 1 and δ(n) = 0 if n > 1.
• For each f for which f(1) ≠ 0 there exists a g such that f * g = δ. g is called the Dirichlet inverse of f.
• f * g = g * f (commutativity)
• f * (g + h) = f * g + f * h = (g + h) * f (distributivity)
• If f is completely multiplicative then f (g*h) = (f g)*(f h).
• 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.)
• In particular, every multiplicative f has a Dirichlet inverse g that is also multiplicative.

With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity δ, 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 and consequently, the arithmetic functions f with f(1) ≠ 0 (the units of the Dirichlet ring) form an abelian group under convolution with identity element δ. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

[edit] Weak multiplicative functions

If are weak multiplicative functions, then so is . That is, weak multiplicative functions have closure in convolution.

[edit] Proof

Let m,n be relatively prime. We wish to prove that .

For , let D_a be the set of divisors of a. For relatively prime m,n, we claim that the function is a bijection from to D_mn. Indeed, for any and , so . Furthermore, for each , there exist unique d_m,d_n such that , , d_md_n = d. Thus p is bijective. As a result of our claim, we have the identity , for any functions a,b mapping subsets of into . In particular, we may let the domains of a and b be D_m,D_n, and define a(d_m) = f(d_m)g(m / d_m) and b(d_n) = f(d_n)g(n / d_n). We then have .

But since each divisor of m is relatively prime to every divisor of n, we have

,

as desired.

[edit] Dirichlet inverse

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

and for n > 1,

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

[edit] Dirichlet series

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

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:

for all s for which both series of the left hand side converge, one of them at least converging absolutely (note that simple convergence of both series of the left hand side DOES NOT imply convergence of the right hand side!). This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.

[edit] References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9