Section24.4Multiplication

Subsection24.4.1Some coincidences

One surprising thing about this is that the Euler products for the Riemann \(\zeta\) function and the Möbius function's series are simply multiplicative inverses. That is, \[\prod_p \frac{1}{1-p^{-s}}=1/\left(\prod_p 1-p^{-s}\right)\, .\] We can check this numerically as well; in the following examples, we use \(s=2\).

Hey, at least they agree up to quite a few digits when we approximate it, so that is a start at reasonability!

Remark24.4.1

Zeta has interesting values at integers. Recall from our exploration of the average value of \(\sigma\) that \(\zeta(2)=\frac{\pi^2}{6}\) (though before we just used this as a sum, and didn't call it \(\zeta(2)\)).

Though Euler calculated many even values of \(\zeta\), which all looked like \(\pi^{2n}\) times a rational number, it was only in 1978 that \(\zeta(3)\) was shown to be irrational. It was then named after the man who proved this, Roger Apéry (so it is called Apéry's constant).

To this day, it is only known that at least one of the next four odd values (\(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\)) is irrational.

Subsection24.4.2Multiplication of both kinds

Let's reinterpret this just a little bit. Assuming we can prove that all this makes sense (which we haven't, yet), we have the following:

The arithmetic functions \(u\) and \(\mu\) are inverses, that is \(u\star \mu=I\).
The Dirichlet series of these functions are also inverses. \[\prod_p \frac{1}{1-p^{-s}}=1/\left(\prod_p 1-p^{-s}\right)\, .\] So \[\sum_{n=1}^\infty\frac{\mu(n)}{n^s}=1/\zeta(s)\]

This is all not a coincidence.

Theorem24.4.2

Take the following notation:

Take \(f(n)\) and \(g(n)\) to be two arithmetic functions.
Let \(h=f\star g\) be their Dirichlet product.
let \(F,G,H\) be the corresponding Dirichlet series (in \(s\)).

Then if the series \(F\) and \(G\) converge absolutely for any \(s\) you care about (not necessarily all \(s\)), then \(H=FG\) for those \(s\) as well.

Proof

This is really a quite remarkable and deep connection between the discrete/algebraic point of view and the analytic/calculus point of view.