Evaluation of Riemann-Stieltjes integral in Laurent expansion of zeta function I'm probably being really stupid but in a proof of the Laurent expansion of the Riemann zeta function the quantity \begin{equation} S_r(t) = \sum_{n \leq t} \frac{(\log (x/n))^r}{n} \end{equation} is used in the following integral \begin{equation} \int_1 ^\infty \frac{1}{t^s}dS_r(t) = \frac{r!}{s^r}\int_1 ^\infty \frac{1}{t^s}dS_0(t) = \frac{r!}{s^r}\sum_{n=1} ^\infty \frac{1}{n^{r+1}}. \end{equation} The first equality is fine (and I've left out a few steps) but it's the second I'm having issues with. Obviously $S_0(t) = \sum_{n \leq t} \frac{1}{n} = \log t + \gamma +O\left(\frac{1}{x}\right)$ and I'm assuming that's where the second equality comes from but I'm not as au fait as I'd like to be with R-S integration and can't quite fathom the steps.

Ah, don't worry, I forgot about the fact that $S_0(x) - S_0(x-1) = \frac{1}{[x]}$ and so we just use the summatary characterisation.