TITLE: fractional part of Riemann zeta $\sum_{s=2}^\infty \{\zeta (s)\}=1$
QUESTION [6 upvotes]: $$\sum_{s=2}^\infty \{\zeta (s)\}=1$$
where $\zeta (s)$ is Riemann zeta, $\{x\}$ denotes the fractional part of the real number $x$


The problem was proposed by Michael Th. Rassias
$\{\zeta(2)\}=\frac{\pi^2}6-1,$ How to go on?
Thanks a lot!

REPLY [3 votes]: Here is how you prove it.
$$ s_n = \sum_{s=2}^{n}\left\{ \zeta(s)\right\} = \sum_{s=2}^{n} (\zeta(s) - \lfloor \zeta(s)\rfloor ) = \sum_{s = 2}^{n} (\zeta(s) - 1 ) $$
$$ \implies \lim_{n\to \infty}s_n = 1. $$
Notes:
1) 

$$\left\{ x\right\} = x - \lfloor x\rfloor, $$

where $\lfloor x\rfloor$ is the floor function.
2) 

$$ \lfloor \zeta(s)\rfloor = 1,\quad \forall s\geq 2. $$

3) $$ \sum_{s=2}^{\infty} (\zeta(s) - 1 ) =1. $$.