The following is Section 3.4. of Titchmarsh's book The Theory of the Riemann Zeta-Function:
My questions are on the last paragraph:
1- How does $1+ai$ being a zero of $\zeta(s)$ imply $1-ai$ being a zero of $\zeta(s)$ as well?
2- Let $1+ai, 1-ai$ be zeros of $\zeta(s)$, then we have $\zeta(s) = (s-1-ai) g_1(s)$ and $\zeta(s) = (s-1+ai) g_2(s)$. On the other hand, $\zeta(s) = \dfrac{1}{s-1}+g_3(s)$. Then how $\lim_{s \to 1} \dfrac{\zeta^2(s)\zeta(s+ai)\zeta(s+ai)}{\zeta(2s)} < \infty$. Just because two zeros times two infinity is finite does not look rigorous.
3- In Theorem 11.1. of Apostol's book Introduction to Analytic Number Theory existence of a real number $\sigma_a$ as the abscissa of absolute convergence (i.e. a number such that if the function converges for that point then it converges all points in the halp plane contains that number) is proved when the function was represented by Dirichlet series. How to prove that for analytic continuation of Dirichlet series? In particular, in this paragraph, why divergence of $f(s)$ at $\sigma_0=-1$ implies abscissa of convergence is the line $\sigma_0=-1$?
3- In the last sentence, how Eq. 3.4.1 gives $f(\frac12) \ge 1$ and how $f(\frac12)=0$?
1: $\zeta$ is conjugate invariant so $\zeta (\bar z)=\overline {\zeta(z)}$ hence if $z$ root then $\bar z$ root too.
2: The local form of an analytic function says that if $h$ has a double pole and $g$ a double zero at some point $w$, then $hg$ is analytic at $w$ since near $w$ we have $h=h_1/(z-w)^2, g=(z-w)^2g_1(z), h_1, g_1$ analytic and then $hg=h_1g_1$ analytic near and at $w$ (here take $w=1, h(z)=\zeta^2(z), g(z)=\zeta(z+ai)\zeta(z-ai)$)
3: Landau's Theorem says that the abscissa of convergence of a Dirichlet series with nonnegative coefficients is a singularity of the respective analytic function; here this means that the abscissa must be a singularity of $$f(s)=\dfrac{\zeta^2(s)\zeta(s+ai)\zeta(s-ai)}{\zeta(2s)}$$ as the coefficients are nonnegative
But since $1$ is regular and the numerator has no other real singularities, it follows that the singularity must be a real zero of $\zeta (2s)$ and the first such is $-1$, hence $\sigma_0 \le -1$ so $f(s)$ is given by its Dirichlet series on $\Re s >-1$; plugging $s=1/2$ and noting that the coefficients are non negative and start with $1$ we get $f(1/2) \ge 1$
But $\zeta(2\times 1/2)=\infty$ and the numerator is finite at $1/2$ so $f(1/2)=0$ (finite/infinite) which is the required contradiction!