# Interesting zeta-like series

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.

• 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 $\lim_{\epsilon \to 0} k_\epsilon^{-1} \zeta(1+\epsilon) =\lim_{\epsilon \to 0}\frac{k_\epsilon^{-1} }{\epsilon} < \infty$ – reuns Nov 13 '18 at 18:41
• 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. – AlephNull Nov 13 '18 at 20:10