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 ?


1 Answer 1


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

  • $\begingroup$ 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 $\endgroup$ Commented Sep 2, 2015 at 16:27
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    $\begingroup$ Why did you denote it like this? Is there some relation to the prime counting function I am missing? $\endgroup$
    – user9072
    Commented Sep 2, 2015 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 $\endgroup$ Commented Sep 2, 2015 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 $\endgroup$ Commented Sep 2, 2015 at 16:44
  • $\begingroup$ Thanks for the clarification. I can follow the reasoning. $\endgroup$
    – user9072
    Commented Sep 2, 2015 at 16:59

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