@Pietro Majer Not for the moment. I am now trying to put in order my second arXiv paper arXiv:1702.06487. Adding some new material to it. Your question appear to be interesting. I will try.

The transformation for $\beta^5$ is equation (7.3) p. 150 of W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005) 137-162. In pag. 154 of the same paper it is found the one for $\epsilon^2$.

Iwaniec treat the analytic continuation in Chapter 6 of his book.

But Iwaniec do not use zeta at Re s=1 only for $\sigma>1$. Well it is true he uses the behavior near $\sigma=1$ but he appear to prove this in page 10 again without using $\sigma=1$.

He uses "semi-elementary methods" to prove $M(x)\ll x (\log x)^{-A}$. From this I think there is no much difficulty. Essentially this is done by Landau when he proves $\sum \mu(n) \log n/n=-1$. (Primzhalen p. 612-613)

@user1952009 I have localized (T/pi) log(T/2pi)-T/pi in the line sigma=1. That these are all need a work. That the missing zeros are at most O( log T) is easy. I think it can be proved that there is at most a finite number off sigma=1.

@user1952009 I have localized (T/pi) log(T/2pi)-T/pi in the line sigma=1.

@user1952009 But it is easy to obtain asymptotic expansion for vertical lines far of the critical strip. I said I will not give the details.

@Marius Overholt the equality $$\sum_{n=1}^\infty (1+\mu(n))e^{-nt}=\frac{e^{-t}}{1-e^{-t}}+e^{-t}$$ is not true. In what sense do you mean this?

@Franz Lemmermeyer may you give an explicit reference of Dirichlet claim, please.