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I am learning more about the theory of additive functions ($f(nm)=f(m)+f(n)$) and I am struck by how powerful the theorems given are. For example, we have a complete characterization of not only what the means of sequences are, but the frequency with which they appear in every interval (a limiting distribution) as given for functions where this distribution exists (Erdős-Winter theorem) and for functions where we need some sort of normalization (Erdős-Kac theorem).

Where more is there to go? What questions do modern-day mathematicians have about additive functions? I'd wager that we want to look at the analytic continuations of their Dirichlet series (i.e zeros of the prime zeta function), but I really am not sure.

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    $\begingroup$ Proving versions of the Erdős–Kac theorem in other settings is still a popular hobby, see for instance the survey article Granville, Soundarajan - Sieving and the Erdős–Kac theorem. $\endgroup$
    – Daniel Loughran
    Commented Oct 26, 2020 at 16:29
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    $\begingroup$ There are also some more recent papers on this topic: arxiv.org/abs/2001.10970, arxiv.org/abs/1711.08396 $\endgroup$
    – Daniel Loughran
    Commented Oct 26, 2020 at 16:30
  • $\begingroup$ Hmm.. Interesting. I will make sure to read these papers. $\endgroup$
    – Milo Moses
    Commented Oct 26, 2020 at 16:31

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