= Subpower bound for the divisor function
{title2=$\tau(n)\ll_\eta n^\eta\quad(\eta>0)$}
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>, $a+1\le2^a\le p^{\eta a}$ for every $a\ge0$. For each of the finitely many smaller <primes>, $\sup_{a\ge0}(a+1)p^{-\eta a}$ is finite. Multiplying these finitely many constants proves the bound. Products of a fixed number of divisor functions obey the same conclusion after reducing $\eta$.
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