I mean the Laurent series at $s=1$.
I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$,
based on the integral formula given in Wikipedia. But I cannot solve this integral except by using Mathematica. Tried complex analytic ways but no luck. Any suggestions? Thanks for your attention!
On
We can show that $$ \zeta(s)=\frac1{1 - 2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}\tag{1} $$ converges for $\mathrm{Re}(s)>0$ by applying the Generalized Dirichlet Test. To apply the test, we need to show that the partial sum of $(-1)^{n-1}$ is bounded, $|n^{-s}|\to0$, and $$ \sum_{n=1}^\infty\left|n^{-s}-(n+1)^{-s}\right|\tag{2} $$ converges.
When $s\in\mathbb{R}$, $n^{-s}$ head straight from $1$ to $0$, so the sum in $(2)$ is $1$. However, if $s\not\in\mathbb{R}$, then $n^{-s}$ spirals into $0$, and it is not immediately obvious that that spiral has finite length.
Let's look at how $n^{-s}$ spirals into $0$:
$\hspace{6pt}$(a)$\hspace{6pt}$ $\arg(n^{-s}) = -\log(n)\mathrm{Im}(s)$
$\hspace{6pt}$(b)$\hspace{6pt}$ $|n^{-s}| = n^{-\mathrm{Re}(s)} = e^{-\log(n)\mathrm{Re}(s)}$
Thus, $n^{-s}$ lies on the spiral $r = e^{t\theta}$ where the constant $t = \mathrm{Re}(s)/\mathrm{Im}(s)$. The length of this curve from $r=1$ to $r=0$ is easily computed to be $|s|/\mathrm{Re}(s)$. Thus, the total variation of $n^{-s}$, as given in $(2)$, is bounded by $|s|/\mathrm{Re}(s)$, and therefore, the sum in $(1)$ converges.
Consider $(1)$ to first order in $s-1$. $$ \frac1{1 - 2^{1-s}}=\frac1{s-1}\frac1{\log(2)}+\frac12+O(s-1)\tag{3} $$ and $$ \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}=\log(2)+(s-1)\sum_{n=1}^\infty(-1)^n\frac{\log(n)}{n}+O\left((s-1)^2\right)\tag{4} $$ Therefore, $$ \zeta(s)=\frac1{s-1}+\frac{\log(2)}{2}+\frac1{\log(2)}\sum_{n=1}^\infty(-1)^n\frac{\log(n)}{n}+O(s-1)\tag{5} $$ Next, we can use the Euler-Maclaurin Sum Formula to compute $$ \sum_{k=1}^n\frac{\log(k)}{k}=C+\frac{\log(n)^2}{2}+O\left(\frac{\log(n)}{n}\right)\tag{6} $$ and $$ \sum_{k=1}^n\frac1{k}=\log(n)+\gamma+O\left(\frac1n\right)\tag{7} $$ Then, we can use $(6)$ and $(7)$ to get $$ \begin{align} \sum_{n=1}^\infty(-1)^n\frac{\log(n)}{n} &=-\lim_{n\to\infty}\left(\sum_{k=1}^{2n}\frac{\log(k)}{k}-2\sum_{k=1}^n\frac{\log(2k)}{2k}\right)\\ &=\small-\lim_{n\to\infty}\left(\left(C+\frac{\log(2n)^2}{2}\right)-\left(C+\frac{\log(n)^2}{2}\right)-\log(2)(\log(n)+\gamma)\right)\\ &=\gamma\log(2)-\frac{\log(2)^2}{2}\tag{8} \end{align} $$ Combining $(5)$ and $(8)$ yields $$ \zeta(s)=\frac1{s-1}+\gamma+O(s-1)\tag{9} $$
There is a nice proof in Titchmarsh's "The Theory of the Riemann Zeta Function": putting $\,[x]=\,$ the greatest integer not greater than $\,x\in\mathbb R\,$, we get: $$\lim_{x\to 1^+}\left[\zeta(s)-\frac{1}{s-1}\right]\stackrel{\mathbf{(7)}}=\int_1^\infty\frac{[x]-x+\frac{1}{2}}{x^2}\,dx+\frac{1}{2}=\\=\int_1^\infty\frac{[x]-x}{x^2}+\frac{1}{2}\int_1^\infty\frac{dx}{x^2}+\frac{1}{2}=\int_1^\infty\frac{[x]-x}{x^2}\,dx+1=$$$$=\lim_{n\to\infty}\left[\sum_{m=1}^{n-1}\left(\int_m^{m+1}\frac{[x]dx}{x^2}-\int_m^{m+1}\frac{dx}{x}\right)+1\right]=$$$$\lim_{n\to\infty}\left[\sum_{m=1}^{n-1}m\left(\int_m^{m+1}\frac{dx}{x^2}\right)-\log n+1\right]=$$$$=\lim_{n\to\infty}\left[\left(1-\frac{1}{2}+1-\frac{2}{3}+...+1-\frac{m-1}{m}\right)+1-\log n\right]=$$$$=\lim_{n\to\infty}\left(\sum_{m=1}^{n-1}\frac{1}{m+1}+1-\log n\right)=\lim_{n\to\infty}\left(\sum_{m=1}^n\frac{1}{m}-\log n\right)=:\gamma$$
Finally, since we know $\,\displaystyle{\lim_{s\to 1^+}(s-1)\zeta(s)=1}\,$, we have that $\,s=1\,$ is a simple pole of $\,\zeta(s)\,$ with residue $\,1\,$, so the above gives the free coefficient of the Laurent expansion of $\,\zeta(s)\,$ around $\,1$
Proof of (7): We use the next form of Abel's summation by parts formula (all the time, $\,n\in\mathbb N\,$):
Lemma: Let $\,\phi(x)\,$ be any function with continuous derivative in $\,[a,b]\,$, then $$\sum_{a< n\leq b}\phi(n)=\int_a^b\phi(x) dx+\int_a^b\left(x-[x]-\frac{1}{2}\right)\phi'(x)dx+\left(a-[a]-\frac{1}{2}\right)\phi(a)-\left(b-[b]-\frac{1}{2}\right)\phi(b)\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\color{blue}{(1)}$$
$\color{red}{\text{Proof}}$: Taking $\,a=n\,,\,b=n+1\,$ and integrating by parts we get at once: $$\int_n^{n+1}\left(x-n-\frac{1}{2}\right)\phi'(x)dx=\left.\left(x-n-\frac{1}{2}\right)\phi(x)\right|_n^{n+1}-\int_n^{n+1}\phi(x)dx=$$ $$=\frac{1}{2}\left(\phi(n)+\phi(n+1)\right)-\int_n^{n+1}\phi(x)dx\Longrightarrow$$ $$\Longrightarrow \int_n^{n+1}\phi(x)dx+\int^{n+1}_n\left(x-[x]-\frac{1}{2}\right)\phi'(x)dx+\left(n-[n]-\frac{1}{2}\right)\phi(n)-\left(n+1-[n+1]-\frac{1}{2}\right)\phi(n+1)=\phi(n+1)=\sum_{n<m\leq n+1}\phi(m)\,\,,\,\,m\in\mathbb N$$
which shows both that the formula works for the above particular case and that it's enough to check for the case $\,n\leq a<b\leq n+1\,$ , so again integrating by parts: $$\int_a^b\left(x-n-\frac{1}{2}\right)\phi'(x)dx=\left(b-n-\frac{1}{2}\right)\phi(b)-\left(a-n-\frac{1}{2}\right)\phi(a)-\int_a^b\phi(x)dx$$ Comparing with the equality promised by the lemma, we see the RHS of $\,(1)\,$ above reduces to $$-\left(b-[b]-\frac{1}{2}\right)\phi(b)+\left(b-n-\frac{1}{2}\right)\phi(b)=\left([b]-n\right)\phi(b)$$ and this equals zero unless $\,b=n+1\,$, but then the last expression above equals $\,\phi(b)=\phi(n+1)\,$, which is the LHS in $\,(1)\,\;\;\;\;\;\;\;\square$
Now $\,(7)\,$ follows from the above with $$a_n=1\,\,,\,\forall n\in\mathbb N\,\,,\,\,\phi(n):=n^{-s}\quad\text{ and }\quad\,\,A(x):=\sum_{0\leq n\leq x}a_n=[x]$$ and we get
$$\zeta(s):=\sum_{n=1}^\infty\frac{1}{n^s}=\sum_{n=1}^\infty a_n\phi(n)=s\int_1^\infty\frac{[x]dx}{x^{1+s}}$$
Finally, we just write $$\frac{1}{s-1}=\int_1^\infty\frac{dx}{x^s}$$