Introduction to analytic number theory

What Hacker News says

1. susamJul 2023
   Although I have read many textbooks, I have not read too many textbooks for each subject, so it is quite difficult to write a list of "best textbooks" for each subject. Time is limited. So I have to make the best out of whichever textbooks I do get the time to read. Having said that, these are some of the textbooks that have had significant influence on me and broadened my understanding of the abstract or concrete world around me.

   1) Computer System Architecture (3rd ed.): This book was phenomenal in my life because it taught me exactly where the hardware meets the software. This book helped me to understand exactly how the hardware fetches, decodes and executes the software on a physical electronic circuit. In fact, I liked this book so much that I implemented the theoretical machine described in the book (called Mano Machine by many) on a Xilinx XC9572 PC84 CPLD Trainer kit. I did not make any assembler or compiler for it, so I had to push DIP switches on the kit to load instructions on a RAM (implemented on another kit) to program the CPU and test it.

   2) The C Programming Language (2nd ed.) by Brian Kernighan and Dennis Ritchie: This is the first programming language book that I read cover to cover. It was quite formative in my journey of learning to program the computer.

   3) Introduction to Analytic Number Theory by Tom M. Apostol: I found this to be a fascinating textbook on analytic number theory. The book begins with simple properties of divisbility but it then soon introduces several new concepts like the Möbius function, Dirichlet product, the prime number theorem, etc. The book exposes various subtle nuances of the Riemann zeta function with great rigour and thoroughness. A complete analytic proof of the prime number theorem is also presented in this book.

   4) Concrete Mathematics: A Foundation for Computer Science (2nd ed.) by Ronald Graham, Donald Knuth, and Oren Patashnik: Great collection of problems that are insightful and entertaining.

   5) Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter: Not exactly a textbook but it is a very intriguing book with a large assortment of interesting concepts related to formal systems, paradoxes, self-reference, isomorphism, meaning of "meaning", etc. all packaged neatly into a single book.

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2. MrBraJan 2015
   "An overview of Inter-universal Teichmüller Theory and [...] how we can begin to understand this theory":

   http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Int...

   Also, from an anonymous post on 4chan @ http://boards.4chan.org/sci/thread/6931488/are-you-ready-for... :

   "How to understand IUTeich from the bottom up:

   - Algebra (Israel Gelfand)

   - The Method of Coordinates (Israel Gelfand)

   - How to Prove It: A Structured Approach (Daniel Velleman)

   - Kiselev's Geometry - Planimetry and Stereometry

   - Trigonometry (Israel Gelfand)

   - What Is Mathematics? An Elementary Approach to Ideas and Methods (Richard Courant)

   - A Course of Pure Mathematics (G.H. Hardy)

   - Linear Algebra (Kenneth Hoffman and Ray Kunze)

   - Elementary Differential Equations (William Boyce and Richard DiPrima)

   - Topology (James Munkres)

   - Calculus On Manifolds (Michael Spivak)

   - Principles of Mathematical Analysis (Walter Rudin)

   - Real and Complex Analysis (Walter Rudin)

   - Functional Analysis (Walter Rudin)

   - Partial Differential Equations (Lawrence Evans)

   - Analysis On Manifolds (James Munkres)

   - Abstract Algebra (David Dummit and Richard Foote)

   - Algebraic Topology (Allen Hatcher)

   - Introduction to Smooth Manifolds (John Lee)

   - Foundations of Differentiable Manifolds and Lie Groups - (Frank Warner)

   - Galois Theory (Harold Edwards)

   - Linear Representations of Finite Groups (Jean-Pierre Serre and Leonhard Scott)

   - A Classical Introduction to Modern Number Theory (Kenneth Ireland and Michael Rosen)

   - Introduction to Analytic Number Theory (Tom Apostol)

   - Modular Functions and Dirichlet Series in Number Theory (Tom Apostol)

   - Riemannian geometry (Peter Petersen)

   - The Theory of the Riemann Zeta-Function (Edward Charles Titchmarsh)

   - An Introduction to Teichmüller Spaces (Yoichi Imayoshi and Masahiko Taniguchi)

   - A Course in p-adic Analysis (Alain Robert)

   - Foundations of p-Adic Teichmuller Theory (Shinichi Mochizuki)

   - Mochizuki's papers at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.htm... "

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3. tzsJun 2025
   It depends on the field.

   For example in programming base e is more common. For example log is base e in C/C++, JavaScript, Java, Python, Perl, Mathematica, Fortran, and Rust.

   Another example is number theory. I just checked a few number theory books from my library and most used base e: Hardy & Wright's "An Introduction to the Theory of Numbers", Apostol's "Introduction to Analytic Number Theory", Baker's "A Comprehensive Course in Number Theory", Ingham's "The Distribution of Prime Numbers", Baker's "Transcendental Number Theory", Ribenboim's "The Book of Prime Number Records", Kumanduri & Romero's "Number Theory with Computer Applications", and Niven's "Irrational Numbers".

   The only number theory book I found using ln rather than log was Gelfond's "Transcendental & Algebraic Numbers".

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4. turtleyachtMar 2025
   Author references Introduction to Analytic Number Theory.

   Link: https://archive.org/details/introductiontoan00apos_0

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5. susamJul 2024
   My own form of learning in public has been to host book clubs for the books I have been reading and then taking notes, sharing insights about them through blog posts, discussing them on IRC channels, etc. I have done two such book clubs so far. The first one was for Introduction to Analytic Number Theory (Apostol, 1976) and the second one was for Mastering Emacs (Petersen, 2022).

   I have the meeting logs, details, etc. for them available here: https://susam.net/cc/

   I plan to begin one more on Galois Theory (Stewart, 2015) quite soon. I am just waiting for the hardcover book to arrive! If this type of thing interests you, I'd love for you to join the discussion either in the IRC channel or the Matrix room linked on the page above.

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6. susamJul 2024
   The first book club I organised focussed on "Introduction to Analytic Number Theory" by Apostol (1976). While the book excelled in rigour, some members, especially those without a strong mathematics background, found the constant alternation between theorems and proofs a bit too dry. Personally, I thoroughly enjoyed it. See <https://susam.net/journey-to-prime-number-theorem.html> for a related post.

   For the current book club, I've chosen something more lightweight with a more relaxed writing style. Although I'm slightly concerned about the level of rigour in this new book, I'll be able to assess it better as we make more progress through the book.

   Rudin is also on my mind. But maybe that's for a future series of book club meetings!

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