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.