I try to find a represantation for the derivative of the riemann zeta function. I do have for $\Re(s)>0$ and $s \neq 1$
$\zeta(s)=\dfrac{1}{s-1}+1-s\int_{1}^{\infty} \dfrac{x-\lfloor x \rfloor}{x^{s+1}} dx$
My question is this expression true:
$\zeta(s)' =- \dfrac{1}{(s-1)^2}-\int_{1}^{\infty} \dfrac{x-\lfloor x \rfloor}{x^{s+1}} dx+s(s+1)\int_{1}^{\infty} \dfrac{x-\lfloor x \rfloor} {x^{s+2}} dx$
feeling I have made a mistake, assuming it is true we obtain
$\lim_{t \rightarrow 0} \zeta(1+it)'(it)+\zeta(1+it)= 1-\int_1^{\infty} \dfrac{x-\lfloor x \rfloor}{x^{2}} dx$