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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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!

  • 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.

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.

Since I can’t find a table of contents online, the chapters are:

  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.

Besides Apostol, try

  • A Primer of Analytic Number Theory: From Pythagoras to Riemann,
    by Stopple.

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

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.

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.

Maybe you can look into the book:- Analytic Number Theory:Exploring the anatomy of integers by Florian Luca. It is a very introductory book in Analytic Number Theory and deals with a lot of beautiful examples.