I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order to develop a narrower path for self study.
In addition, it would be helpful to know if there is a book that does a good job showing off how the
complex analysis machinery can be used effectively in number theory,
or at least one with a good amount of well-developed examples in order to provide a wide background of the tools that complex analysis gives in number theory.
Best Answer
This question is like asking how abstract algebra is useful in number theory: lots of it is used in certain areas of the subject so there's no tidy answer. You probably won't be using Morera's theorem directly in number theory, but most of single-variable complex analysis is needed if you want to understand basic ideas in analytic number theory. A few topics you should pay attention to are: the residue theorem, the argument principle, the maximum modulus principle, infinite product factorizations (esp. the Hadamard factorization theorem), the Fourier transform and Fourier inversion, the Gamma function (know its poles and their residues), and elliptic functions. Basically pay attention to the whole course! There really isn't a whole lot in a first course on complex variables where one can say "that you should ignore if you are interested in number theory".
If you want to be careful and not just wave your hands, you need to know conditions that guarantee the convergence of series and products of analytic functions (and that the limit is analytic), the existence of a logarithm of an analytic function (it's not the composite of the three letters "log" and your function), that let you reorder terms in series and products, that justify termwise integration, and of course the workhorse of analysis: how to make good estimates.