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Let $\{u_k\}_{k\in\mathbb{N}}$ be an increasing (or even nondecreasing?) sequence of positive reals, limiting to infinity.

For the series $f(s) := \sum_{k=1}^{\infty} (u_k)^{-s}$ with $s$ a complex variable, by the results stated in the comments and answers here, we know there exists $\sigma \in [-\infty,+\infty]$ such that the convegence set in $\mathbb{C}$ of $f(s)$ is $\{\mathrm{Re}(s) > \sigma\}$.

I am also wondering about the following:

  1. Is $f(s)$ holomorphic with a unique continuation to a meromorphic function on $\mathbb{C}$?

    If yes, then:

  2. If $\sigma\notin\{\pm\infty\}$, does the continuation of $f(s)$ have a single pole located exactly at $\sigma +0i \in \mathbb{C}$?

For example, for $u_k = k$ we get $f$ as the Riemann zeta function, and the answers to the above questions are yes, with $\sigma = 1$.

For background, I am thinking about this in the context of the above series being the zeta function of an elliptic differential operator. After eqn (1.20) in this paper, the author claims that if $u_k = \lambda_k$ is the (ordered) spectrum of a positive elliptic differential operator in dimension $n$, then $\sigma \leq n/2$.


Edit: removed "analytic" from the continuation.

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  • $\begingroup$ Note that if $u_k=e^{p_k}$ where $p_k$ is the $k$ the prime the series (under the obvious substitution) becomes a power series with natural boundary unit circle, so as Dirichlet series the abscissa $\Re s =1$ is natural boundary $\endgroup$
    – Conrad
    Commented Jul 5, 2022 at 0:12

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In general, Dirichlet series do not have meromorphic (much less analytic) continuations to $\mathbb C$ (even ignoring the common pole(s) at $s=1$).

Googling "Estermann phenomenon", you will find out that Estermann showed in 1928 that many natural-looking Dirichlet series, even with Euler products, have demonstrable natural boundaries. E.g., $\sum_n d(n)^3/n^s$, where $d(n)$ is the number of positive divisors of $n$.

Also, various people (maybe including Harald Bohr) had earlier shown that Dirichlet series with "random" coefficients (in various sense) should be expected to have natural boundaries.

Kurokawa c. 1985 expanded Estermann's results to prove non-continuability of an even wider class of otherwise natural Dirichlet series.

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  • $\begingroup$ Thank you for this example and explanation! However, I think the example series you are using is not of the type I described in my question. (Since $d(n)^3 n^{-s}$ can't be written as $u_n^{-s}$, for arbitrary $s$, for some $u_n$ depending only on $n$.) $\endgroup$ Commented Jul 4, 2022 at 19:00
  • $\begingroup$ @IndraneelTambe2, well, yes, that example I could remember is not exactly of the type you discuss. On the other hand, I can remember it. :) And it's a data-point. Harald Bohr's (and others') work does address "generalized Dirichlet series" in the sense you want, but I do not recall much of that. $\endgroup$ Commented Jul 4, 2022 at 19:02

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