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I'm wondering about a possible extension of Ikehara's theorem for Dirichlet series with coefficients that are not necessarily positive. Consider the following:

Let $D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s}$ with:

  1. $a(n) = O(n^\epsilon)$ for all $\epsilon > 0$.
  2. $\sum_{n \leq x} a(n) \ll x^c$ for some $c > 0$.
  3. $D(s)$ has meromorphic continuation to $\Re(s) \geq c$ with a simple pole at $s = c$, with residue $R$.

Question: Does this imply $$\sum_{n \leq x} a(n) \sim \frac{R}{c} x^c \quad \text{as} \quad x \to \infty \text{ ?}$$ If we apply Theorem 8.1 (Korevaar, Tauberian theory, Ch. III) to $D(s+c-1)$ we get this result, but with stronger conditions:

  1. $a(n) = O(1)$.
  2. $D(s)$ is analytic for $\Re(s) > c$.
  3. Same meromorphic continuation as above.

I'm not sure it holds with my weaker hypotheses. Any ideas are welcome - maybe a way to prove it, or a counterexample, or similar work you know. If you saw similar extensions or have thoughts on more conditions we might need...

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  • $\begingroup$ I am not convinced you are correct about the application of the theorem in Korevaar as you need $a(n)n^{1-c}=O(1)$ which is a stronger condition for $0<c<1$ than $a(n)=O(1)$ $\endgroup$
    – Conrad
    Commented Aug 29 at 22:41
  • $\begingroup$ Thank you for pointing that out. In fact, I have: $\sum_{n \leq x} a_n \ll x^{c+\epsilon}$ hence $\sum_{n \leq x} a_n n^{c-1} \ll x^{2c-1+\epsilon}$ for $c \neq \frac{1}{2}$. I am thus more specifically interested in the case where $\frac{1}{2} < c < 1$. The question becomes: if $a_n = O(1)$ and $\sum_{n \leq x} a_n \ll x^{c+\epsilon}$ and $D(s) = \sum_{n \geq 1} a_n n^{-s}$ is analytic in $\Re(z) > c$ and on $\Re(z) = c$ except for $z = c$ which is a simple pole with residue $R$, do we have $$\sum_{n \leq x} a_n \sim (R/c) x^c$$ $\endgroup$
    –  Babar
    Commented Sep 1 at 10:06
  • $\begingroup$ If that were true for example the RH couldn't fail with an isolated simple zero on the $c$ line (since then a translate of $1/\zeta$ satisfies your conditions but the sum of the coefficients oscillates) and that of course is unknown $\endgroup$
    – Conrad
    Commented Sep 1 at 13:38

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