Doubt in a deduction from complex analysis to be uses in analytic number theory

Solution 1:

I believe your confusion in 1 is you are switching variables in your head. We want to show for fixed $a$, $I(s,a)$ is entire in $s$. However, the curves $C_1,C_2$ and $C_3$ are defined in the $z$ variable. So you don't need to show uniform convergence for small disks as a separate case. There is no direct interplay between the disk in $s$ and the curves defined in $z$. So, if you show uniform convergence for arbitrarily large disks in the $s$ variable, you are finished.

Implicitly, the author thinks of $I(s,a)=I_1(s,a)+I_2(s,a)+I_3(s,a)$ where $I_j$ is integrating over $C_j$. To the author , it is clear $I_2$ is entire as the integrand is holomorphic in $z$ in some domain containing $C_2$.

Proving 2 for finite length curves is tantamount to proving you can differentiate under an integral in appropriate circumstances. For curves of infinite length, you need to do some work involving bounds and uniform convergence. (as the author does).

Suppose $f(s,z)$ is defined on $U\times V$ and is holomorphic in either variable while the other is fixed. Let $C$ be a finite length curve in $V$. Then $I(s)=\int_C f(s,z) dz$ is holomorphic and $I'(s)=\int_C f_s(s,z) dz$. Proof: differentiate under the integral sign.