Riemann zeta function

For $\left\{{a_k}\right\}$ a sequence in ${\mathbf{R}}$, define $$\begin{align*} A(t) := \sum_{0\leq k \leq t} a_k \end{align*}$$ For $\phi\in C^1({\mathbf{R}})$ continuously differentiable, \begin{align*} \sum_{x

Fix $s\in {\mathbf{C}}, a_{k} = 1$ for $k\geq 1$, and $\phi(x) := x^{-s}$. Then $A(t) = {\left\lfloor t \right\rfloor}$, and $$\begin{align*} \sum_{n=1}^{\lfloor x\rfloor} \frac{1}{n^{s}}=\frac{\lfloor x\rfloor}{x^{s}}+s \int_{1}^{x} \frac{\lfloor u\rfloor}{u^{1+s}} \,du \end{align*}$$ Now take $\Re(s) > 1$ and $\lim_{x\to\infty}$ to yield $$\begin{align*} \zeta(s)=s \int_{1}^{\infty} \frac{\lfloor u\rfloor}{u^{1+s}} \,du \end{align*}$$ Use this to derive Dirichlet’s theorem.md): \zeta(s) has a simple pole at s=1 with $\mathop{\mathrm{Res}}_ \zeta(s: $/zeta(s)$ has a simple pole at $s=1$ with $/Res_{s=1} /zeta(s) = 1$. This works for other Dirichlet series.