I want to prove the existence of the following limit: \begin{equation} \lim_{N \to \infty}\left(\sum_{n=1}^N \frac{1}{n^s} - \int_1^N \frac{dx}{x^{s}}\right) \end{equation}
for $s \in \mathbb{C}$ with Re$(s) = 1 $? I tried to argue via Cauchy criterion to prove that \begin{equation} \left|\sum_{n=M}^N \frac{1}{n^s} -\int_M^N \frac{dx}{x^s}\right| \end{equation} goes to $0$, as $N,M \to \infty$. I believe \begin{equation} \left|\frac{1}{n^s} -\int_n^{n+1}\frac{dx}{x^s}\right| \end{equation} is bounded by something like $\frac{1}{n^{\alpha}}$ where $\alpha>1$ but really could not prove it. Can someone help me with this? Thanks!
Using Riemann–Stieltjes integration (with $\{x\}=x-\lfloor x\rfloor$ denoting the fractional part of $x$) and integrating by parts, \begin{align*} \sum_{n=1}^N \frac{1}{n^s} - \int_1^N \frac{dx}{x^{s}} &= 1 + \int_1^N \frac{d\lfloor x\rfloor}{x^s} - \int_1^N \frac{dx}{x^{s}} \\ &= 1 - \int_1^N \frac{d\{x\}}{x^{s}} \\ &= 1 - \frac{\{x\}}{x^{s}} \bigg|_1^N -s \int_1^N \frac{\{x\}}{x^{s+1}} \,dx = 1 - \frac{\{N\}}{N^{s}} -s \int_1^N \frac{\{x\}}{x^{s+1}}. \end{align*} The right-hand side converges as $N\to\infty$ for any $s\in\Bbb C$ with $\Re s >0$ (because $0\le\{x\}<1$ and the integral converges absolutely for those $s$). Therefore for any such $s$, we have a convergent limit $$ \lim_{N\to\infty} \bigg( \sum_{n=1}^N \frac{1}{n^s} - \int_1^N \frac{dx}{x^{s}} \bigg) = 1 -s \int_1^\infty \frac{\{x\}}{x^{s+1}}. $$