For $s \in ]0;1[$, the Riemann serie $\displaystyle \sum_{k>0}\frac{1}{k^s}$ diverges and we have the more precise asymptotic expansion : $$\sum_{k=1}^n\frac{1}{k^s} = \frac{n^{1-s}}{1-s}+ \zeta(s)+ \underset{n\to+\infty}{o}(1)$$
where $\zeta$ is the analytic continuation of the Riemann zeta function to $\mathbb{C}-\left\{1\right\}$.
I was wondering :
For $s\le 0$, is there an analogous formula ? By analogous, i mean something that relates the behavior of the Riemann sum $\displaystyle \sum_{k=1}^{n}k^{-s}$ when $n\to +\infty$ to the value of $\zeta(s)$ ?
For $0<s<1$, what can be said about the following terms in the asymptotic development above ? Are they related to other "well-known" functions ?
Thanks,
