Truncation of a divisor weight by its second moment
= Truncation of a divisor weight by its second moment
If $\sum_{n\leq N}\tau_k(n)^2\ll N(\log N)^{O(1)}$, then for any $X\geq1$,
$$
\sum_{\substack{n\leq N\\\tau_k(n)\geq X}}\tau_k(n)
\leq X^{-1}\sum_{n\leq N}\tau_k(n)^2
\ll\frac{N(\log N)^{O(1)}}X.
$$
This separates a divisor-weighted exponential sum into a bounded-weight part and a sparse large-weight part.