I am studying analytic number theory from Tom M Apostol introduction to analytic number theory,
Actually Concept of analytic continuation was not taught by my instructor who took complex analysis course. So I began self studying it for Complex Variables with Applications By Ponnusamy and Silvermann. But I doesn't feel confident in some theorems or exercises of Analytic number theory which uses analytic continuation.
Here is an example where Apostol uses analytic continuation ( In last line of proof) .
Equation (17) is $\zeta(s) = e^{G(s) } $
What I can think about it --> In equation (17) both RHS and LHS are analytic if s is real and s>1 . But which result Apostol is using to extend its domain of analyticity. ( is he using the fact that both LHS and RHS are analytic for s>1 even when s belongs to Complex Numbers and hence extension of domain of analyticity is justify. I don't always get a complete understanding when analytic continuation is used .
Can someone please explain it if my argument is wrong.
The heart of the matter is the following theorem.
Identity Theorem. Let $f$ be an analytic function in an open connected set $\Omega\subset\mathbb{C}$, and suppose that the zero set of $f$ has an accumulation point in $\Omega$. Then $f\equiv0$.
By applying this theorem to the difference between LHS and RHS of (17), we infer that the difference is identically zero in $\mathrm{Re}(\sigma)>0$.