Are there bounds in the literature on sums of the form $$\sum_{\omega(n)= k} e(\alpha n) \;\;\;\;\;\text{or}\;\;\;\;\; \sum_{\Omega(n)=k} e(\alpha n)$$ for $\alpha$ on minor arcs (i.e., not very close to a rational with small denominator)? It's not hard to prove such bounds given things that are standard and available, but I do not want to reinvent the wheel.
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asked Jan 25, 2021 at 10:21
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$\begingroup$ It's so interesting. I'm not sure whether the best bounds for the sums would be obtained using sieve methods, power series methods, or Dirichlet series methods. They just seem so prone to being analyzed. $\endgroup$– Milo MosesJan 29, 2021 at 0:16
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$\begingroup$ M. Radziwill has pointed me towards Tenenbaum, "Facteurs premiers de sommes d'entiers". $\endgroup$– H A HelfgottFeb 16, 2021 at 23:02
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$\begingroup$ Tenenbaum's article has an answer. One can give a sharper estimate by supplementing his procedure with a contour integration as in Selberg-Delange. $\endgroup$– H A HelfgottFeb 21, 2021 at 8:32
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$\begingroup$ Ah, interesting. I couldn't get over the fact that $e(\alpha n)$ is not multiplicative with respect to $n$, however, so that didn't get me anywhere. $\endgroup$– Milo MosesFeb 21, 2021 at 17:38
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