number theory which is close to analysis

You could try the short book by Hugh Montgomery, which focuses closely on the interactions of harmonic analysis and number theory.

Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis by Hugh L. Montgomery Series: CBMS Regional Conference Series in Mathematics (Book 84) Paperback: 220 pages Publisher: American Mathematical Society (October 11, 1994) ISBN-10: 0821807374


It seems like you want to discover analytic number theory. There is a lot of it. A good comprehensive modern book I would recommend is

Iwaniec, Kowalski - Analytic number theory.

Example areas with applications of harmonic analysis include the circle method and modular forms.


In an other fashion, you can be interested in how Fourier analysis (series decompositions, Poisson formula) is fundamental in :

  • Trace formulas (kind of generalization of Poisson formula in the non-real-and-commutative case)
  • Computing functional equations for zêta-functions and reaching Tamagawa numbers (those are volumes of fundamental quotient spaces in adelic settings)
  • Modular and automorphic forms

For trace formulas and automorphic forms, I would say that an efficient and pleasant first lecture is H. Iwaniec, Spectral Methods of Automorphic Forms, AMS. In order to see how Fourier analysis works well in those settings, you can read Tate's thesis, it is the GL(1) case, available in Cassels-Frohlich or in Lang, Algebraic Number Theory, Springer GTM.

For Tamagawa numbers, the book of Vignéras, Arithmétique des algèbres de quaternions, Springer LNM, is a very nice reference. It is more or less translated in Reid-MacLachlan, The arithmetic of Hyperbolic 3-Manifolds, Springer GTM.

Hoping you could uncover those lovely topics ;)