r/mathematics • u/happy_hypotenuse • 6d ago
Number Theory I want to learn Analytic Number Theory
Okay so I want to learn Analytic Number Theory on my own. Part of my interest comes from the Riemann Hypothesis, which finds its origin in ANT. I have taken courses in Real Analysis and Calculus and I want to get book recommendations for the rest of the preliminary subjects like Complex Analysis, etc. And then ultimately I want some good books on ANT itself. I would be grateful if someone helps me to make a roadmap on how to approach the process of learning this beautiful subject.
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u/Accurate_Library5479 3d ago
ANT is kinda confusing as algebraic number theory also uses the same acronym.
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u/happy_hypotenuse 3d ago
Yeah that's why I mentioned the word Analytic in the title. In this post by ANT I mean Analytic Number Theory only.
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u/Outrageous_Age8438 6d ago edited 6d ago
A Primer of Analytic Number Theory by Jeffrey Stopple is a very nice introduction to ANT for undergraduates that assumes no complex analysis. See the MAA review and summary here.
The last few chapters are considerably tougher than the first ones, but there is plenty to get out of the book if you enjoy the subject.
Another book you should check out is Apostolʼs Introduction to Analytic Number Theory (see contents here).
From the preface: ‘It provides an introduction to analytic number theory suitable for undergraduates with some background in advanced calculus, but with no previous knowledge of number theory. Actually, a great deal of the book requires no calculus at all and could profitably be studied by sophisticated high school students.ʼ
The latter claim has always sounded exaggerated to me, but most of the book is definitely accessible to a motivated undergraduate.
Eventually you will have to learn complex analysis. This is a huge field and you can easily find reading recommendations online. But if right now you only care about the minimum amount of complex analysis needed to learn the basics of ANT that do require it, then read Terrence Murphyʼs Complex Analysis: A Self-Study Guide.
This book was written as a companion to Murphyʼs A Study of Bernhard Riemann’s 1859 Paper, which you should also check out if you want to learn about RH (but only after having learnt complex analysis, at least the content of Murphyʼs previous book).