2
$\begingroup$

For the $\sigma$ function, the ratio $\sigma(m)/m$ is known as the abundancy index. Is there any special name for $\phi(m)/m$ with $\phi$ the Euler's totient function ?

$\endgroup$
1
$\begingroup$

I called it $\pi^{-1}(m)$ in a number theory article I posted on the ArXiv. I did not scour the literature, but I conjecture that Erdos never came up with a name for it (he didn't in the papers I saw that talked about the quantity), and I would also like to know if anyone else did.

Gerhard "No Need For A Link" Paseman, 2015.09.02

$\endgroup$
  • $\begingroup$ I also did it in the Westzynthius question that I occasionally promote. I wrote a manifesto in which I suggested the quantity deserved more attention, and then decided I had gone too far, and kept myself from posting it online. Gerhard "Doesn't Mean I Was Wrong" Paseman, 2015.09.02 $\endgroup$ – Gerhard Paseman Sep 2 '15 at 16:27
  • 1
    $\begingroup$ Why did you denote it like this? Is there some relation to the prime counting function I am missing? $\endgroup$ – user9072 Sep 2 '15 at 16:32
  • $\begingroup$ Different context. I used sigma ^ -1 for the sum of reciprocals of distinct prime divisors, and pi ^ -1 for product of (1 - reciprocal). I was dealing with counting consecutive nontotatives (non coprimes), not primes. Gerhard "Seemed Natural At The Time" Paseman, 2015.09.02 $\endgroup$ – Gerhard Paseman Sep 2 '15 at 16:42
  • $\begingroup$ Also, while there may be a relation between $\pi(x)$ and $\pi^{-1}(m)$, I never planned to use the two in the same article. Gerhard "Like Edward Teller And Anti-Teller" Paseman, 2015.09.02 $\endgroup$ – Gerhard Paseman Sep 2 '15 at 16:44
  • $\begingroup$ Thanks for the clarification. I can follow the reasoning. $\endgroup$ – user9072 Sep 2 '15 at 16:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.