# A good reference to begin analytic number theory

I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about analytic number theory. I'd like to know if there would be a book that I could find (or notes from a teacher online) that would introduce me to analytic number theory's classical results. Any suggestions?

Thanks for the tips,

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This question came to mind when I read math.stackexchange.com/questions/153013/… ... I realized I wanted to read such a book for a long time. – Patrick Da Silva Jun 2 '12 at 20:24
I am doing a course now in analytical number theory (math.stanford.edu/~ksound/Math155Spr12/Math155.html) and I follow the book by Apostol. I believe it is a great book and the exercises are great. I have a soft copy of the book and I can email it to you if you want. – user17762 Jun 2 '12 at 21:33
@DylanMoreland I got it already. I'm looking forward to reading it. It is very up to date (2003) so that's really a pro. – Pedro Tamaroff Jun 2 '12 at 23:43
@PatrickDaSilva I have a digital "distributable" version =) – Pedro Tamaroff Jun 3 '12 at 0:56
@PatrickDaSilva Thanks again. Probably in its time and in a class context, Apostol has its place. But after checking out the first few pages of Davenport, it looks much better. – Andrew Jun 7 '13 at 21:24

I'm quite partial to Apostol's books, and although I haven't read them (yet) his analytic number theory books have an excellent reputation.

Introduction to Analytic Number Theory (Difficult undergraduate level)

Modular Functions and Dirichlet Series in Number Theory (can be considered a continuation of the book above)

I absolutely plan to read them in the future, but I'm going through some of his other books right now.

Ram Murty's Problems in Analytic Number Theory is stellar as it has a ton of problems to work out!

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I'll see if some other people suggest other things and check the best answer at the end. But I appreciate your answer! I did not know this book before. – Patrick Da Silva Jun 2 '12 at 20:56
@PatrickDaSilva I'm self-studying from Apostol's (it's not that difficult) Eugene is studying from it too, but in a ANT course, and we sometimes meet up in the chat to discuss some of its proposed problems. Feel free to step by! – Pedro Tamaroff Jun 2 '12 at 21:45
@Peter Tamaroff : Agreed! Where do you discuss such things? – Patrick Da Silva Jun 2 '12 at 21:56
@PatrickDaSilva Here – Pedro Tamaroff Jun 2 '12 at 21:57
@Michael Boratko : I've read a few pages of Apostol, I like his algebraic point of view on multiplicative functions, an algebraic mind like me understand things quickly :) but in overall I like the book! Thanks for the reference =) – Patrick Da Silva Jun 3 '12 at 3:28
• If you haven't read the chapter on Dirichlet's theorem on primes in arithmetic proression in Serre's Course in arithmetic, I highly recommend that you do. You can read it independently of what came before.

• I liked the book of Ayoub when I was a student. My memory is that it is somewhere between a textbook and a monograph, and that it covers lots of fundamental topics, such as partitions, Dirichelt's theorem, the circle method, and so on. I found it compelling enough that I failed an English course because I spent all my time reading the book instead of writing the required essay.

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I just finished a reading course with Chandrasekharan's Introduction to Analytic Number Theory and I really enjoyed it.

It starts at the basics, estimating the size of the $n^{\text{th}}$ prime using Euclid's proof of the infinitude of primes, and follows a logical path starting there and ending at the prime number theorem. I think of the road chosen as the "scenic route"; the journey is just as important as the goal.

Along the way the author's enthusiasm is tangible as he takes detours to touch on interesting results and makes it a point to showcase a large variety of problems and techniques. When proving theorems he'll often opt for a proof given by someone other than the original author, and once or twice he includes multiple proofs which illustrate different perspectives. And, when introducing definitions, they are never just tools to be filed away for later use; they are always placed in the context of an interesting problem and given respect on their own.

1. The unique factorization theorem
2. Congruences
3. Rational approximation of irrationals and Hurwitz's theorem
4. Quadratic residues and the representation of a number as a sum of four squares
5. The law of quadratic reciprocity
6. Arithmetical functions and lattice points
7. Chebyshev's theorem on the distribution of prime numbers
8. Weyl's theorem on uniform distribution and Kronecker's theorem
9. Minkowski's theorem on lattice points in convex sets
10. Dirichlet's theorem on primes in an arithmetical progression
11. The prime number theorem

He also avoids functional equations completely, which I appreciate.

I found a couple reviews online here and here.

I've answered a couple questions using material from the book here and here

I just want to mention that I really, really dislike Apostol's book. It's incredibly dry and thoroughly uninspiring. I found reading the proofs to be a chore, whereas the proofs are the juciest part of Chandrasekharan. To me, Apostol is not a book to be "read" or learned from. It's decent as a reference.

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Besides Apostol, try

• Analytic Number Theory: An Introductory Course, by Bateman and Diamond.

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Dear lhf, I gave +1 for Stopple. I looks quite pleasant. Since you mentioned them both, I was wondering how it compares with Apostol. Is it as rigorous? Is Apostol as accessible? Does Apostol cover substantially more material (seems that way from the Tables of Contents)? Thanks very much, – Andrew May 21 '13 at 11:09
@Andrew, Stopple is more elementary than Apostol but just as rigorous and as enjoyable. Apostol covers more ground: for instance, he proves Bertrand's postulate, Dirichlet's Theorem on primes in arithmetic progressions, and the prime number theorem. – lhf May 21 '13 at 12:23
Thanks very much – Andrew May 21 '13 at 13:38

Well, I am highly biased with the book "Analytic Number Theory" by Iwaniec and Kowalski. It is a wonderful book to learn thoroughly. However, Apostol's book is also pretty good for beginning.

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While parts of the book by Iwaniec and Kowalski can be read by a beginner, I think one needs considerable experience to grasp the totality of it. For example, I think conceptually it is difficult for a beginner to appreciate the point of the chapter about bilinear forms. – blabler Jul 2 '13 at 16:58

I looked at loads of books when I started studying analytic number theory and for me the best by far was Jameson's "The Prime Number Theorem". Even though it's mainly about the prime number theorem, it goes into all the basics too. Apostol's "Introduction to Analytic Number Theory" is also good.

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