Subpower bound for the divisor function ($\tau (n){\ll }_{\eta }{n}^{\eta }(\eta >0)$)

For each $\eta >0$, the divisor function satisfies the displayed bound. Write $n=\prod p^{a_p}$, so $\tau (n)=\prod (a_p+1)$. For all sufficiently large primes, $a+1\le 2^a\le p^{\eta a}$ for every $a\ge 0$. For each of the finitely many smaller primes, ${\sup }_{a\ge 0}(a+1)p^{-\eta a}$ is finite. Multiplying these finitely many constants proves the bound. Products of a fixed number of divisor functions obey the same conclusion after reducing $\eta$.