# Series of reciprocal of integers

This is a question I asked myself today...



Do you know if it is possible to build a strictly-increasing sequence $$(u_n)_{n\in\mathbb{N}^\star}$$ of positive integers such that $$\displaystyle\sum_{n=1}^{+\infty}\frac{1}{u_n}<+\infty$$ and such that for any given $$0<\varepsilon<1$$, one has $$\displaystyle\sum_{n=1}^{+\infty}\frac{1}{u_n^\varepsilon}=+\infty$$ ?



This is looking for a Dirichlet series (with reciprocal of $$u_n$$ coefficients) with abscissa of convergence $$\sigma=1$$ and that has a finite limit at $$s=1$$, does that exist ? It would mean that the corresponding L-function (if the series was extendable around $$s=1$$) would have no pole at $$s=1$$. This is tough because the L-function would not be in the Selberg class, and those are hard to study...

You can take $$u_n = \lceil n \log^2 n\rceil$$: $$\sum_{n=1}^\infty \frac{1}{n \log^2 n} < \infty$$ but $$\sum_{n=1}^\infty \frac{1}{n^\varepsilon \log^{2\varepsilon} n} = \infty$$ for every $$\varepsilon < 1$$.
• Oh that's totally true ! I tried $u_n=n\log(n)$ (the PNT makes sure $p_n$ works), but I didn't try that ! Thanks ! – Anthony Dec 7 '18 at 2:01