The following is Section 3.6. of Titchmarsh's book The Theory of the Riemann Zeta-Function:
My questions are:
1- Eq. 3.6.1 holds for $\sigma >1$. Eq. 3.6.2 holds for $\sigma > 1 - A/\log t$. There are two things about the green-underlined equation: (a) The second term must be $-A_2|\sigma -1| \log^2 t$ and the absolute value operator comes from triangle inequity. (b) This equation is a result comes from combining Eqs. 3.6.1 and 3.6.2, so it holds for $\sigma > 1$, the intersection of two domains. Holding this for $\sigma >1$ and not for $\sigma > 1 - A/\log t$ ruins the original purpose of Titschman which was to show that there is some narrow rectangular with vertices $1 \pm \epsilon(t) + it_0 $ and $1 \pm \epsilon(t) + it $ where $\zeta(s)$ is non-zero. Am I right by these two remarks?
2- I suppose the correct inequality for the red-underlined one is $1 - A/\log t < \sigma <1$ (a typo). (a) To get Eq. 3.6.3 we supposed $\sigma -1 = \log^{-9} t$ (so $\sigma > 1$) and Eq. 3.6.4 is a result from Eq. 3.6.3 but the common domain for the two is empty! (b) The correct form of Eq. 3.6.4 is $$|\zeta(\sigma+it)| = |\zeta(\sigma+it) - \zeta(1+it) + \zeta(1+it)| > |\zeta(1+it)| - |\zeta(\sigma+it) - \zeta(1+it)| > A \log^{-7} t - B (1- \sigma) \log^{2} t,$$ where $B$ is the positive constant from the $O$ of Eq. 3.6.2. But $A=B$ is considered! Am I right by these two remarks?
3- $\dfrac{\zeta'(s)}{\zeta(s)} = \dfrac{d}{ds} \log \zeta(s)$, i.e. the differentiation is done on $s$ and not on the real part of $s$. So Eq. 3.6.7 is incorrect. Can this proof for estimating $\log \zeta(s)$ be adjusted and if so how? Otherwise, how to prove $\log \zeta(s) = O( \log^9 t)$?