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Respected All.

I am studying number theory where I came to know that $\varphi(n), \sigma(n)$ both are multiplicative function ; In other words, if $(m,n)=1$ then \begin{align} \sigma(mn)=\sigma(m)\sigma(n),\\ \varphi(mn)=\varphi(m)\varphi(n) \end{align}

But my question is: what if $(m,n)\neq 1$ ? In this case, how shall I relate $\varphi(mn), \varphi(m), \varphi(n)$ ? Is there any closed expression ? What about $\sigma(mn), \sigma(m), \sigma(n)?$ Well I know the equality will not hold, but if $\varphi(mn)=\lambda \varphi(m)\varphi(n)$ what should I write for $\lambda$ ? Same for $\sigma$.

Please help me. Thanks in advance

Martin Sleziak
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KON3
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  • See [this question](http://math.stackexchange.com/questions/114841/proof-of-a-formula-involving-eulers-totient-function) and [other questions linked there](http://math.stackexchange.com/questions/linked/114841) for the part about totient functions. – Martin Sleziak Aug 12 '15 at 09:31
  • The book Sandor, Mitrinovic, Crstici: *Handbook of number theory I* contains [section called on $\sigma(mn)$](https://books.google.com/books?id=XT1-HjeXFgYC&pg=PA87). But the authors only list some inequalities in this section. – Martin Sleziak Aug 12 '15 at 09:36

1 Answers1

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$\varphi(mn)=\varphi(m)\varphi(n)\dfrac{d}{\varphi(d)}$ where $d=\text{gcd}(m,n)$

ZFR
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