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I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis.

Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(probably), I will have some thing to work on(I mean, chance of discovering something new)?

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  • $\begingroup$ Maybe this question should be changed to something like: "Application of Fourier Analysis in Number Theory". $\endgroup$
    – user40023
    Commented Mar 29, 2015 at 9:12
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    $\begingroup$ You say nothing at all about your background in number theory. Have you studied it at all? $\endgroup$
    – KConrad
    Commented Mar 29, 2015 at 14:00
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    $\begingroup$ Oh, if you are currently a PhD student then speak to faculty in your department. Do any of them work in number theory? $\endgroup$
    – KConrad
    Commented Mar 29, 2015 at 15:43
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    $\begingroup$ Theta function is both in number theory and harmonic analysis. en.wikipedia.org/wiki/Theta_function $\endgroup$
    – Mohammad Ghiasi
    Commented Mar 29, 2015 at 16:32
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    $\begingroup$ Compulsory literature: claymath.org/sites/default/files/ezeta.pdf $\endgroup$
    – Count Iblis
    Commented Mar 29, 2015 at 19:59

7 Answers 7

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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

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    $\begingroup$ I second this. I thought of the same book, and I am pleased I am not alone! $\endgroup$
    – GH from MO
    Commented Apr 1, 2015 at 18:07
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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.

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    $\begingroup$ I think this book is pretty tough going for a beginner. $\endgroup$
    – Greg Martin
    Commented Apr 9, 2015 at 17:09
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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 ;)

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  • $\begingroup$ There are more accessible applications of Fourier analysis in number theory, starting from some of the fast multiplication algorithms and evaluations of Gauss sums. $\endgroup$
    – Noam D. Elkies
    Commented Sep 25, 2016 at 4:56
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An older book that I really liked was Ayoub's An Introduction to the Analytic Theory of Numbers. It describes several famous classical theorems proved by analytical methods. It requires very little background beyond analysis. You can read the introduction here.

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An area where analysis (especially the Fourier kind) is used quite heavily is in the study of so-called "Beurling-Selberg maximal functions," which have applications to several areas in number theory, such as counting lattice points and studying the behavior of zeta- and $L$-functions. The basic idea is that there are some naturally occurring functions, such as characteristic functions of intervals, balls, etc., which obviously don't have nice analytic properties, and it can be very fruitful to find functions which majorize and minorize your "bad" function that have "nice" properties with respect to their fourier transforms.

The paper entitled

A survey on Beurling-Selberg majorants and some consequences of the Riemann hypothesis

by Emanuel Carneiro seems like it would be a good starting point, but I can't seem to find a full text online. Carneiro and others have written quite a few papers on this topic, such as this one:

https://www.ndsu.edu/pubweb/~littmann/research/Gaussian_2_4.pdf

by Carneiro, Littmann, and Vaaler, which I would describe as mostly analysis, with some discussion of applications to number theory. You might find this stuff interesting.

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The fourth volume of Stein & Shakarchi's series on analysis has a nice account of the application of harmonic analysis to lattice point problems (e.g. Gauss' circle problem and the Dirichlet divisor problem).

Martin Huxley's "Area, Lattice Points, and Exponential Sums" is a more thorough (but very readable) account of the business of counting lattice points.

My advisor (Alex Iosevich) has done work (with others) in this area, using some more recent techniques in harmonic analysis (or as Alex would say "using nuclear weapons on small animals"). Anyway, it's still an active area, though as suggested above, you might want to work on whatever is popular in your department.

Also, Vinogradov's "Elements of Number Theory" has a lot of nice exercises on exponential sums. Recently, techniques from additive combinatorics have led to new results on exponential sums; Bourgain wrote a nice survey called "Sum-product Theorems and Applications".

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I think it is remarkable that nobody mentioned André Weil's Basic Number Theory until now. André Weil made both marvellous contributions to harmonic analysis on locally compact Abelian groups and to number theory.

In Basic Number Theory, familiarity with number theory is not a prerequisite. However, the reader is expected to be familiar with the basic theory of locally compact Abelian groups and the Haar measure on such groups since these methods are extensively used to prove results in number theory. I think this approach is non-standard, but it beautifully shows the application of harmonic analysis to number theory.

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