Dirichlet Inverse Problem

**VinceW** · June 27th 2011, 02:16 PM

The Dirichlet Convolution is this:

$(f * g)(n) = \sum\limits_{d \mid n} f(d)g(\frac{n}{d})$

The $\iota(n)$ function:

$\iota(n) = \begin{cases} 1 & \text{if } n = 1 \\ 0 & \text{if } n > 1\\ \end{cases}$

The arithmetic function $g$ is said to be the inverse of the arithmetic function $f$ if $f * g = g * f = \iota$ . Show
that the arithmetic function $f$ has an inverse if and only if $f(1) \neq 0$ . Show that if $f$ has an inverse it is
unique. (Hint: When $f(1) \neq 0$ , find the inverse $f^{-1}$ of $f$ by calculating $f^{-1}(n)$ recursively, using the
fact that $\iota(n) = \sum\limits_{d \mid n} f(d) f^{-1}(\frac{n}{d})$ .)

I'm really stumped on how to approach this one. Obviously the inverse doesn't exist when $f(1) = 0$ , but beyond that, how do I tackle this one?

**chisigma** · June 27th 2011, 10:14 PM

By definition, given an arithmetical function $f(n)$ with $f(1) \ne 0$ , its inverse $f^{-1} (n)$ obeys to the relation...

$f^{-1} * f = \sum_{d|n} f^{-1} (d)\ f(\frac{n}{d}) = \iota(n)$ (1)

For n=1 the (1) gives us...

$f^{-1}(1)\ f(1) = 1 \implies f^{-1}(1) = \frac{1}{f(1)}}$ (2)

For n>1 we have...

$\sum_{d|n} f^{-1} (d)\ f(\frac{n}{d}) = f(1)\ f^{-1} (n) + \sum_{d|n, d<n} f^{-1} (d)\ f(\frac{n}{d}) = 0$ (3)

... so that from (2) and (3) we derive...

$f^{-1} (n) = - \frac{1}{f(1)}\ \sum_{d|n, d<n} f^{-1} (d)\ f(\frac{n}{d})$ (4)

... and this formula permits us to compute the $f^{-1}(n)$ recursively...

Kind regards

$\chi$ $\sigma$

Thread: Dirichlet Inverse Problem

LinkBack

Thread Tools

Display

Dirichlet Inverse Problem

Re: Dirichlet Inverse Problem

Similar Math Help Forum Discussions

Two questions concerning Laplacian (and a Dirichlet Problem)

Dirichlet Problem-Partial Differential Equations

Dirichlet problem for a rectangle

Dirichlet Divisor problem

An extension of Dirichlet's Divisor Problem