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Let $FI(N)$ denote the product of all $\phi(N)$ [relatively prime numbers $<N$] . And define $SFI(N)$ as the product of remaining $N-\phi(N)$ numbers $\le N$ (Which are not relatively prime to $N$)

Then I have the following questions

Q-1 For what values of $N$, $FI(N)<SFI(N) ?$
Q-2 For what values of $N$, $d[FI(N)]<d[SFI(N)] ?$ Where $d(n)$ is number of divisors.

Q-3 Are there asymptotics for $FI(N)$ and $SFI(N)?$

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  • $\begingroup$ @IgorRivin What about $SFI(N)$? $\endgroup$
    – Shivam Patel
    Commented May 19, 2014 at 3:52
  • $\begingroup$ Since $SFI(N) \times FI(N) = N!,$ I am not sure what you mean... $\endgroup$
    – Igor Rivin
    Commented May 19, 2014 at 4:06
  • $\begingroup$ @IgorRivin I understood your point . Now can you please tell me about asymptotics of a functions $A(N)$ which is the sum of all ϕ(N) and $B(N)$ the sum of remaining N−ϕ(N) ? $\endgroup$
    – Shivam Patel
    Commented May 19, 2014 at 4:21
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    $\begingroup$ The asymptotics of $\sum_1^N\phi(n)$ would be a good question for math.stackexchange.com, provided it doesn't duplicate a question already asked there. But it is also answered in many introductory Number Theory textbooks, and probably on a great many websites, as well. $\endgroup$
    – Gerry Myerson
    Commented May 19, 2014 at 5:30

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