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So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level.

He has studied analytic number theory from several books, among them are Hardy’s Introduction to the Theory of Numbers, Apostol’s Analytic Number Theory and Modular Forms and Dirichlet series and Rosen’s Classical Introduction to Modern Number Theory.

Besides that, he has a general graduate background in analysis - measure theory, functional analysis, probability theory and complex analysis, along with some smooth manifolds and differential geometry.

My question is as follows - what texts would you recommend for someone like this to go further in analytic number theory? He is also interested in the more analytic aspects of algebraic number theory - he has mentioned Tate’s thesis as something he would like to study. However he has no particular end goal in mind.

Thanks in advance!

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    $\begingroup$ There are many texts past the basic level. Montgomery-Vaughan, Iwaniec-Kowalski, Titchmarsh just to name a few. $\endgroup$
    – Wojowu
    Commented May 4, 2022 at 10:37
  • $\begingroup$ Thank you for the recommendations. Will pass these along to him! $\endgroup$
    – Nate River
    Commented May 4, 2022 at 10:43
  • $\begingroup$ Another text is Cohen's two part series "Number Theory, Volume 1: Tools and Diophantine Equations" and "Number Theory, Volume 2: Analytic and Modern Tools." $\endgroup$
    – Joshua Stucky
    Commented May 5, 2022 at 5:37
  • $\begingroup$ For "analytic aspects of algebraic number theory", Narkiewicz's book is great (e.g. he gives a careful proof of the Chebotarev density theorem). $\endgroup$
    – David Loeffler
    Commented May 6, 2022 at 10:23
  • $\begingroup$ Halberstam & Richert's Sieve Theory published in 1974 is another good one if you want to get into some classical additive problems related to primes. $\endgroup$
    – TravorLZH
    Commented May 8, 2022 at 13:47

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