Details in the proof of irrationality of $\zeta(3)$ I am reading through Wadim Zudilin's lecture notes (https://www.dropbox.com/s/r1972uagq3zb4mc/AMiNT.pdf?dl=0 pages 75-76) and the author assumes some basic real/complex analysis facts without proofs which are the following (in question formats) :
1- Consider the function $g(y) = \Re f(z_0 + iy)$, that is, the real part of $f(z)$ on the contour of integration. How $\frac{d}{dy} g(y)=−\Im \frac{df}{dz}|_{z=z_0+iy}$ is true?
2- How the only $y$ for which $\Im\ln (y^{-2}-1)=0$ is $y=0$?
3- If $r_n^{1/n}$ tends to a positive quantity as $n\to\infty$, how do we conclude that $r_n$ does not vanish for all $n$ sufficiently large?
4- If $\lim_{n\to\infty}|a_n^{1/n}|\lt1$ then how do we know that $|a_n|<1$ for all $n$ suitably large?
 A: For the point (1):
Under the assumptions of the theorem for the Cauchy-Riemann equations if $f(x+iy) = u(x,y) + iv(x,y)$ and $\frac{df}{dz}\Big|_{z=z_{0}+iy}$ exists, then
$$ \frac{df}{dz} \Big|_{z=z_{0}+iy} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} -i \frac{\partial u}{\partial y}$$
Now if $$\displaystyle g(y) = \Re f(z_{0}+iy) = u(z_{0},y) $$
then $$ \frac{dg(y)}{dy} = \frac{\partial u}{\partial y} $$
On the other hand
$$\Im \frac{df}{dz} \Big|_{z=z_{0}+iy} = -\frac{\partial u}{\partial y}$$
$$\Longrightarrow \frac{dg(y)}{dy} = \frac{\partial u}{\partial y} = - \Im \frac{df}{dz} \Big|_{z=z_{0}+iy} $$
In the book $z_{0}$ is assumed $\displaystyle \frac{1}{\sqrt{2}}$
For the point (2):
You have a typo, in the book the author put $\ln(z^{-2}-1)$
Suppose $z = x+iy$
Then
$$z^{-2}-1 = -\frac{(x+iy)^2-1}{(x+iy)^2} = -\frac{\left[(x+iy)^2-1\right](x-iy)^2}{(x+iy)^2(x-iy)^2} = \frac{-y^4-2x^2y^2+x^2-y^2-y^4}{(x^2+y^2)^2} - i\frac{2xy}{(x^2+y^2)^2 }$$
Hence
$$\Im (z^{-2}-1)  = -\frac{2xy}{(x^2+y^2)^2 }$$
$ \Longrightarrow \theta = \operatorname{Arg}(z^{-2}-1) = \Im \ln(z^{-2}-1)=0 $ if
$$ -\frac{2xy}{(x^2+y^2)^2} =0$$
which only happens when $y=0$ given that in the book $x\neq 0$
For the point (4):
This is the ratio test for series:
Suppose that $\lim_{n \to \infty} |a_{n}|^{\frac{1}{n}}$ exists and is strictly less than $1$. Then $\sum_{n=1}^{\infty} a_{n}$ converges absolutely.
Then as a part of the Cauchy criterion we can show that if
$$ \sum_{n=1}^{\infty} a_{n} \;\textrm{ converges then } \;  a_{n} \to 0$$
Then $\forall \epsilon>0 \;\; \exists N\in \mathbb{N}$ such that $n\geq N$ implies that
$$|a_{n}-0|<\epsilon$$
Choose $\epsilon =1$
