While studying Analytic number theory from Tom M Apostol Introduction to analytic number theory I have a doubt in proof of theorem 12.3 .
Proof is – 
I have doubts in first paragraph of proof.
Why doesn't Apostol mentions that uniform convergence is to be proved for a compact disk of |s|$\leq \epsilon $ where this compact disk lies in $C_2$ as it is proved for $C_1$ and $C_3$ . Then how can author be sure that I(s, a) would be analytic in $C_2$ .
Apostol then writes as integrand is entire function of s this will prove that I(s, a) is entire.
If integrand is entire then how does it proves that I(s, a) is entire. Can someone please prove it.
Can someone please explain these 2 doubts.
Best Answer
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.