Lagmental Vicfred

Dirichlet Convolution Multiplies over Divisor Splittings by Vicfred

tags: mathematics, arithmetic functions, Dirichlet convolution, Mobius inversion

The convolution of f and g sums f(d)g(n/d) over every divisor d of n. A small computation will anchor the general statement before the abstraction takes over.

The mathematical object

Arithmetic functions \(f:\mathbf N\to\mathbf C\) form a commutative ring under Dirichlet convolution. Multiplicative functions are determined by their values on prime powers \(p^k\).

$$ (f*g)(n)=\sum_{d\mid n}f(d)g(n/d) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ f*g=g*f,\qquad(f*g)*h=f*(g*h) $$

One explicit computation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{aligned}(\mathbf1*\mathbf1)(12)&=\sum_{d\mid12}1\\&=\#\{1,2,3,4,6,12\}=6=\tau(12).\end{aligned} $$

Why the identity matters

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad (f*g)(n)=\sum_{d\mid n}f(d)g(n/d),\\[5pt] \mathsf{C}\;&:\quad f*g=g*f,\qquad(f*g)*h=f*(g*h). \end{aligned} $$

Where it can fail

Pointwise multiplication and Dirichlet convolution are different operations. Möbius inversion reverses convolution with the constant-one function, not ordinary multiplication.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f*g=g*f,\qquad(f*g)*h=f*(g*h) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Fri 19 July 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.