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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.

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maybe some analytic number theory would be helpful. For example, Apostol's textbook in UTM series. "Introduction To Analytic Number Theory" – natura Oct 25 '10 at 3:08
It would help to know how much number theory background you have. The analytic class number formula would be fantastic if you have the prerequisites. – Cam McLeman Oct 25 '10 at 3:18
I'm not sure how to measure my background,but I'd say my knowledge comes primarly from Ireland and Rosen "A classical Introduction to number theory". Recently I have been taking a look at some more advanced number theory books (algebraic NT) – Daniel Kohen Oct 25 '10 at 3:35
This might be too basic, as my knowledge of complex analysis and analytic number theory together couldn't fill a thimble --- but as an undergraduate I was really happy with Stein & Shakarchi's Complex Analysis, the second half of which is spent investigating the foothills of analytic number theory. In those foothills, at least, it illustrates handily how complex analytic techniques work. – Eric Peterson Oct 25 '10 at 3:44
You might try picking up Koblitz's 'Introduction to Elliptic Curves and Modular Forms'. Modular forms are extremely fascinating objects number theoretically, but require a certain amount of complex analysis even to define. A few years ago I sat in on an Intro to Modular Forms course, and there was so much complex analysis involved that it wasn't until halfway through the term that it became apparent I was taking a number theory course! – Ben Linowitz Oct 25 '10 at 3:47
up vote 19 down vote accepted

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.

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thank's for the answer. I have to add that my intention is not to "ignore things that I'm not interested in", on the contrary, I try to give more time and effort to the topics I think are more "useful", not meaning that I will overread the other concepts. – Daniel Kohen Oct 25 '10 at 13:16
Wanting to give more time to the more useful topics is hard to distinguish from just giving more time to the course overall, since most of it is relevant to number theory. I now thought of something which isn't: Schwarz–Christoffel transformations (explicit ways to identify the upper half-plane with the inside of a polygon). – KConrad Oct 25 '10 at 16:01
Dear KConrad, I want to recall that Schwarz Christoffel mappings is a conceptual approach towards understanding elliptic function theory, since they are the inverses of elliptic functions. This includes Weierstrass $\ws$ and Jacobi $sn$! – Marc Palm Nov 19 '10 at 6:56

I think basic is on the right track. The two big classical theorems in analytic number theory whose classical proofs use some complex analysis are Dirichlet's Theorem on primes in arithmetic progressions and the Prime Number Theorem. (It is also useful to learn about the combination of the two: the Prime Number Theorem for Arithmetic Progressions.)

For the former, I can recommend my own lecture notes:

The second part is explicitly a digested version of the proof Serre presents in his Course in Arithmetic. I don't have a similarly canonical reference to give you for the proof of the Prime Number Theorem (i.e., I don't have any notes on it!), but it can be found in many analytic number theory books, for instance in Apostol's Introduction to Analytic Number Theory, Davenport's Multiplicative Number Theory or G.J.O. Jameson's The Prime Number Theorem.

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For a canonical reference on PNT, perhaps Davenport's "Multiplicative Number Theory"? – Micah Milinovich Oct 25 '10 at 3:46
There is a new book "The Prime Number Theorem" by G.J.O Jameson, which the author says is intended precisely to fill the gap of a book devoted entirely to introducing and proving the PNT.… – Marko Amnell Oct 25 '10 at 4:24
I can vouch for Jameson's book as someone who read, enjoyed, and learnt a huge amount from it. Definitely my first recommendation for anyone interested in the PNT. – Thomas Bloom Oct 25 '10 at 9:37
I agree, Jameson's book is very good, he leads you through the material in a logical way, and doesn't leave things out. Easy to follow. Tom – Tom Dickens Aug 28 '11 at 23:10

I'll second Pete Clark's answer, and note that there are some other big theorems in analytic number theory for which there are proofs using Complex analysis. For example, there is the asymptotic formula for the number of partitions of $n$, which formula is ${e^{\pi\sqrt{2n/3}}\over4\sqrt3n}$. There is a proof in Donald J Newman, Analytic Number Theory, but be warned that the chapter on the partition function is infested with typos.

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The whole of Newman's book was infested by typos, at least the first edition (which I have) was. But there is a second edition, in which I understand most of the typos were corrected (I don't have this). – Robin Chapman Oct 25 '10 at 8:42
Robin, I'm sure you're right - the Partitions chapter is the only one I studied, after having volunteered to give a series of lectures on it. My recollection is that there was a second printing rather than a second edition, and many typos were corrected, but a considerable number remained. – Gerry Myerson Oct 25 '10 at 11:52

And in transcendence theory. The proofs of the Hermite-Lindemann-Weierstrauss Theorem (special case: if $\alpha \neq 0$ is algebraic, then $e^\alpha$ is transcendental) and of the Gelfand-Schneider theorem (special case: if $\alpha\not\in\{0,1\}$ is algebraic, $\beta$ is algebraic and irrational, then $\alpha^\beta$ is transcendental) make fundamental and elegant use of complex analysis.

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Complex analysis is often used in analytic number theory as a tool to evaluate or estimate sums $\sum a_n $ by studying the analytic behaviour (like existence of poles or how fast it grows) of the associated Dirichlet series $\sum a_n n^{-s}$. So for most interesting arithmetical functions (like a_n = number of divisors of n, say), one can study the corresponding Dirichlet series (possibly factor it as a product of various $L$-functions) to obtain information about the sum. One uses the Perron formula or Mellin inversion formula to pass from the sum to a contour integral. Davenport's book is the canonical reference.

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