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The famous Selberg-Delange method takes sequences $a_n$ whose associated DGF $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ has a representation

$$F(s)=G(s;z)\zeta^z(s)$$

where $G(s;z)$ is "nice enough" and turns this into exact asymptotics (in terms of polynomials) on $\sum_{n<x}a_n$. Can the main result of this method be adapted to obtain assymptotics on $\sum_{n<x}\frac{a_n}{n}$ instead?

My overall goal would be to approximate quantities similar to

$$\sum_{\substack{n<x\\\omega(n)=k}}\frac{1}{n}$$

in much the same way as Tenenbaum approximates the quantity $\sum_{\substack{n<x \\ \omega(n)=k}}1$. Is this reasonable to expect? Can these theorems be changed to fit my needs?

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    $\begingroup$ Why not just do partial summation? $\endgroup$
    – alpoge
    Commented Oct 23, 2020 at 7:36
  • $\begingroup$ @alpoge I've tried, but the formulas obtained after applying the S-D method are generally already complex. Adding another layer of summation makes it just that much harder to deal with which is not working in my application, where I use specific facts about the form and properties of these polynomials. $\endgroup$
    – Milo Moses
    Commented Oct 23, 2020 at 15:00
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    $\begingroup$ It is supposedly trivial to get from the asymptotic of $\sum_{n\le x} a_n$ to that of $\sum_{n\le x} a_n/n$ through a partial summation, it is the other way around which is non-trivial, so you are certainly missing something. $\endgroup$
    – reuns
    Commented Apr 16, 2021 at 22:48

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