If $\varphi(mn)=\lambda \varphi(m)\varphi(n)$ what should be written for $\lambda$

Question

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

See this question and other questions linked there 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 $\sigma(mn)$. But the authors only list some inequalities in this section. — Martin Sleziak, Aug 12 '15 at 09:36

score 6 · Accepted Answer · answered Aug 12 '15 at 08:10

6

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

answered Aug 12 '15 at 08:10

RFZ

16,814

what about $\sigma(mn), \sigma(m), \sigma(n)$ in general ? – KON3 Aug 12 '15 at 08:13

If $\varphi(mn)=\lambda \varphi(m)\varphi(n)$ what should be written for $\lambda$

1 Answers1