So this idea was inspired by a post I saw quite a while back asking about the convergence of the series $\sum_{n=1}^\infty \frac{1}{n^{1+|\sin n|}}$ (to which I actually still don’t know the answer).

Consider the zeta-like series $\sum_{n=1}^\infty \frac{1}{n^{s_n}}$, where $s_n$ is a sequence of real numbers with $s_n \geq 1$ and $\displaystyle\liminf_{n \to \infty} s_n = 1$.

My (very) general question is, under what conditions on the sequence $s_n$ does this “$\zeta(s_n)$” converge?

It seems like a difficult question to answer for many such series since it’s somehow wedged between the harmonic series and $p$-series for $p>1$. Any thoughts, even about specific such series, would be interesting to hear.

  • $\begingroup$ You'd need to restrict to a smaller class to propose useful criterion. For $\sum_n n^{-1-|\sin(\pi a n)|}$ the convergence clearly depends on the Diophantine approximation of $a$. With $k_\epsilon$ the least integer such that $|\sin(\pi a k_\epsilon)| \le \epsilon$ you'd get something like $\sum_n n^{-1-|\sin(\pi a n)|}$ converges iff $\lim_{\epsilon \to 0} \sum_n (nk_\epsilon)^{-1-\epsilon} < \infty$ iff $\lim_{\epsilon \to 0} k_\epsilon^{-1} \zeta(1+\epsilon) =\lim_{\epsilon \to 0}\frac{k_\epsilon^{-1} }{\epsilon} < \infty$ $\endgroup$ – reuns Nov 13 '18 at 18:41
  • $\begingroup$ I suppose my question isn’t specific enough, but I am interested in examples of specific classes of s_n for which you can propose a criterion for convergence, like the one you just gave. I am wondering whether it is possible to find some papers about these series, perhaps in a more restricted case. $\endgroup$ – AlephNull Nov 13 '18 at 20:10

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