Consider the following definition of the Dirichlet function which is $1$ at all rational points of the real line and vanishes otherwise:

$$\lim_{j\to\infty}\lim_{k\to\infty} (\cos(k!\pi x))^{2j}.$$

I want to understand why $k!$ is used here instead of just $k,$ which appears to work just as fine, as explained below.

For rational $x=p/q,$ say, as $k\to\infty$ eventually $kx$ (and therefore $k!x$) becomes integral, so that $\cos(kπx)=\pm 1.$ Thus, the limit as $j\to\infty$ is trivially $1$ because of the square. If $x$ is irrational, we must have $|\cos(kπx)|<1,$ so that for any $k,$ we must then have, as $j \to \infty, |\cos(kπx)|\to 0.$

I have not used the operation $!$ applied to the $k$ at all in this alternative definition, which appears to do just as well. Why then is it included in the definition; what am I missing?

  • 3
    $\begingroup$ "eventually $kx$ becomes integral" This is wrong. Take $x=1/2$ for instance - it will be non-integral for all odd $k$. $\endgroup$ – Wojowu Apr 17 at 18:38
  • $\begingroup$ If you were actually doing this on a computer i.e. approximately, the convergence with $k!$ is way faster $\endgroup$ – George Dewhirst Apr 17 at 18:39
  • $\begingroup$ @Wojowu Oh, I'm ashamed. Thanks. $\endgroup$ – Allawonder Apr 17 at 18:39
  • $\begingroup$ @GeorgeDewhirst Of course. I had only been wondering why one couldn't just use $k,$ purely as a valid definition, without other considerations -- like computational efficiency, as you mention. $\endgroup$ – Allawonder Apr 17 at 18:41

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