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This may not exist but I would just like to ask in case.. I think Rudin is probably the best book but it doesn't have any (?) number theory in it.

What is a good textbook (or comprehensive lecture notes) on Fourier analysis which has lots of number theory throughout?

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What exactly do you have in mind? An emphasis on the discrete Fourier transform, for example? – Qiaochu Yuan Apr 16 '11 at 22:05
up vote 7 down vote accepted

Elias Stein and Rami Shakarchi's book Fourier Analysis: An Introduction has a chapter devoted to Dirichlet's Theorem on primes in arithmetic progressions and chapter 7 deals with Fourier analysis on finite abelian groups, so although not entirely with an emphasis in number theory, it has a decent amount I would say, and besides is introductory and elementary in nature.

I'm not sure if you're familiar with this series of books, but in the complex analysis one, which is book II in the series, the authors devote four chapters, namely chapters 6, 7, 9 and 10 to topics which are clearly number theoretic, for example, the Gamma and Zeta functions, the Prime Number Theorem, elliptic functions, Weirstrass $\wp$ function, Eissenstein series, aplications of theta functions to sums of squares, etc.

I also just found these lecture notes which are titled Fourier Analysis in Number Theory and this thesis on Fourier Analytic Applications to Number Theory.

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Perhaps Fourier Analysis on Number Fields ? See a review here.

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This book is about a much more advanced subject (Tate's thesis) and I'm not sure it's what the OP has in mind. – Qiaochu Yuan Apr 16 '11 at 22:04
I'm not sure what the OP has in mind :-) – lhf Apr 16 '11 at 23:30

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