The upper bound of an arithmetic function

Question

Assume that $m=\prod\limits_{i=1}^r{p_i}^{\alpha_i},n=\prod\limits_{i=1}^r{p_i}^{\beta_i}$, where $\alpha_1, \beta_i\in \mathbb{Z}_{\ge0},i=1,2,\cdots,r$, and define that $$t(m,n)=\varphi(m)\varphi(n)(\gcd(m,n))^2\prod_{i=1}^r\left(1+\dfrac{1}{p_i}\right)^{\left\lfloor\frac{\min\{\alpha_i,\beta_i\}}{\max\{\alpha_i,\beta_i\}}\right\rfloor},$$ where $\varphi(n)$ is the Euler totient function, and $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
I've known that $$t(m,n)\ll m^2n^2,$$ but I wonder if there exists a better upper bound of $t(m,n)$ like $(mn)^c$, where $1<c<2$.

Answer 1

When $m=n$ we have $t(m,n) = m^2 \phi(m)^2 \prod_{p\mid m} (1+\frac1p)$ which is $\gg_\varepsilon (m^2)^{2-\varepsilon} = (mn)^{2-\varepsilon}$ for every $\varepsilon>0$; so no better exponent holds in general.