Best Reference Books – Analytic Number Theory

We have compiled the list of Top 10 Best Reference Books on Analytic Number Theory subject. These books are used by students of top universities, institutes and colleges. Here is the full list of top 10 best books on Analytic Number Theory along with reviews.

Kindly note that we have put a lot of effort into researching the best books on Analytic Number Theory subject and came out with a recommended list of top 10 best books. The table below contains the Name of these best books, their authors, publishers and an unbiased review of books on "Analytic Number Theory" as well as links to the Amazon website to directly purchase these books. As an Amazon Associate, we earn from qualifying purchases, but this does not impact our reviews, comparisons, and listing of these top books; the table serves as a ready reckoner list of these best books.

1. “Algebraic Number Theory” by S Lang

“Algebraic Number Theory” Book Review: This book offers an excellent overview of the key theories and outlook of the topic. The book is up to date with the latest development in the field. This book treats cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry. This book is a direct global approach to number fields. This book into topics like Algebraic Integers, Completions, The Different and Discriminant, Cyclotomic Fields, Parallelotopes, The Ideal Function, Adele and Adeles, Elementary Properties of the Zeta Function and L-series, Norm Index Computations, The Artin Symbol, Reciprocity Law, and Class Field Theory, Hecke’s Proof, Functional Equation, Tate’s Thesis, Density of Primes and Tauberian Theorem, The Brauer-Siegel Theorem, Explicit Formulas, etc.

2. “A Course in Arithmetic” by J P Serre

“A Course in Arithmetic” Book Review: This book is resourceful and is easy to understand as it is in simple language and is purely algebraic. The book serves as a classification of quadratic forms over the field of rational numbers. The book covers topics such as Finite Fields, p-Adic Fields, Hilbert Symbol, Quadratic Forms over Q p and over Q, Integral Quadratic Forms with Discriminant ± 1, The Theorem on Arithmetic Progressions, Modular Forms etc.

3. “Introduction to Analytic Number Theory” by T Apostol

“Introduction to Analytic Number Theory” Book Review: This book is the crystallization of the topic to undergraduates irrespective of they have or don’t have any knowledge of the topic earlier. The book starts with the most basic properties of the natural integers. The text in the book is resourceful with enormous information which is well presented and easy to read. The book has a good selection of topics which are easy to understand and are nicely structured with exercises at the end of each chapter. This book covers topics like The Fundamental Theorem of Arithmetic, Arithmetical Functions and Dirichlet Multiplication, Averages of Arithmetical Functions, Some Elementary Theorems on the Distribution of Prime Numbers, Congruences, Finite Abelian Groups and Their Characters, Dirichlet’s Theorem on Primes in Arithmetic Progression, Periodic Arithmetical Functions and Gauss Sums, Quadratic Residues and the Quadratic Reciprocity Law, Primitive Roots, Dirichlet Series and Euler Products, Analytic Proof of the Prime Number Theorem, etc.

4. “Analytic Number Theory” by Henryk Iwaniec and Emmanuel Kowalski

“Analytic Number Theory” Book Review: This book teaches the reader to use the subject to establish results. This book has a vast diversity of concepts and methods. The aim of the book is to show the scope of the theory, both in classical and modern directions, and to exhibit its wealth and prospects, beautiful theorems, and powerful techniques. This book is for graduate students. The book balances the clarity, completeness, and generality. This book has exercises and examples which is intended to improve reader understanding and provide additional information. The reader must have a basic idea and covers topics like Arithmetic functions, Elementary theory of prime numbers, Classical analytic theory of L-functions, Elementary sieve methods, Bilinear forms and the large sieve, Exponential sums, The Dirichlet polynomials, Zero-density estimates, Sums over finite fields, Holomorphic modular forms, Spectral theory of automorphic forms, Sums of Kloosterman sums, Primes in arithmetic progressions, The least prime in an arithmetic progression, The Goldbach problem, Effective bounds for the class number, etc.

5. “A Primer of Analytic Number Theory: From Pythagoras to Riemann” by Jeffrey Stopple

“A Primer of Analytic Number Theory: From Pythagoras to Riemann” Book Review: This is an easy to read book that incorporates all the topics essential for detailed knowledge of the subject. The book is aimed to enhance the analytical skills of the reader in course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. This book shows how the importance of Riemann zeta function and the significance of the Riemann Hypothesis. The book here develops the basic idea of elementary number theory. The book is included with a series of exercises for better understanding and advanced ideas for representation of the topic which is suitable for undergraduates. The book covers topics like Sums and Differences, Products and Divisibility, Order of Magnitude, Averages, Primes, Basel Problem, Euler’s Product, The Riemann Zeta Function, Stirling’s Formula, Explicit Formula, Pell’s Equation, Elliptic Curves, Analytic Theory of Algebraic Numbers, etc.

6. “Analytic Number Theory: Exploring the Anatomy of Integers” by Jean-marie De Koninck and Florian Luca

“Analytic Number Theory: Exploring the Anatomy of Integers” Book Review: This book presents a clear introduction to the subject analytic number theory with a very unique focus on the anatomy of integers, that is, on the study of the multiplicative structure of the integers. The book has a collection of problems at the end of each chapter that have been chosen carefully. The book has answers to solved numericals at the end of each chapter which makes it appropriate for readers who want to test their understanding of the topic. The book covers topic like Prime Numbers and Their Properties, The Riemann Zeta Function, Setting the Stage for the Proof of the Prime Number Theorem, The Proof of the Prime Number Theorem, The Global and Local Behavior of Arithmetic Functions, The Fascinating Euler Function, Smooth Numbers, The Hardy-Ramanujan and Landau Theorems, Sieve Methods, Prime Numbers in Arithmetic Progression, Characters and the Dirichlet Theorem, Selected Applications of Primes in Arithmetic Progression, The Index of Composition of an Integer, Preliminary Notions along with solutions to every problem of the chapters at the end.

7. “Analytic Number Theory: An Introductory Course” by Paul Trevier Bateman and Harold G Diamond

“Analytic Number Theory: An Introductory Course” Book Review: This gives a nice introduction to the topic and is enough for a graduate course.The book is well written and is resourceful. This is an introductory book on the topic and the reader is expected to have a basic idea on real analysis, complex analysis, number theory and abstract algebra.There are various exercises throughout the entire book and at end of the each chapter developments of each particular subject or theorem are given together with references. The book focuses on a collection of powerful methods of analysis that yield deep number theoretical estimates. The book covers topics like Calculus of Arithmetic Functions, Summatory Functions, The Distribution of Prime Numbers, An Elementary Proof of the P.N.T., Dirichlet Series and Mellin Transforms, Inversion Formulas, The Riemann Zeta Function, Primes in Arithmetic Progressions, Applications of Characters, Oscillation Theorems, Sieves, Application of Sieves, etc along with appendices.

8. “Geometric and Analytic Number Theory” by Edmund Hlawka and Johannes Schoißengeier

“Geometric and Analytic Number Theory” Book Review: The book presents the information in a unique way that the reader can not only know how to solve it but also why to solve it. This book includes important findings, examples & exercises. The aim of the book is to present number theory and to introduce basic results from the areas of the geometry of numbers, diophantine approximation, prime number theory, and the asymptotic calculation of number theoretic functions to a beginner. The prerequisite of the book is the reader must have knowledge analytic geometry, differential and integral calculus, together with the elements of complex variable theory, The book covers topics like The Dirichlet Approximation Theorem, The Kronecker Approximation Theorem, Geometry of Numbers, Number Theoretic Functions, The Prime Number Theorem, Characters of Groups of Residues, The Algorithm of Lenstra, Lenstra and Lovász, etc.

9. “Abstract Analytic Number Theory” by John Knopfmacher

“Abstract Analytic Number Theory” Book Review: This book is well written. This book applies classical analytic number theory to a wide variety of mathematical subjects in an arithmetical way. The book deals with the arithmetical semigroups and algebraic enumeration problems, and focuses on arithmetical semigroups with analytical properties of classical type, and gives a deep dive of analytical properties of other arithmetical systems. The book carefully treats fundamental concepts and theorems. The book covers topics like Arithmetical Asymptotic Enumeration, Functions, Enumeration Problems, Arithmetical Semigroups, Arithmetical Semigroups with Analytical Properties of Classical Type and Further Statistical Properties of Arithmetical Functions, The Abstract Prime Number Theorem, Fourier Analysis of Arithmetical Functions, Additive Arithmetical Semigroups, Arithmetical Formations, etc along with appendices and bibliography.

10. “An Introduction to the Theory of Numbers” by G H Hardy and Edward M Wright

“An Introduction to the Theory of Numbers” Book Review: The context here is wrapped up with simple words and with a detailed understanding of all the key topics. The text retains the style and clarity of previous editions making it for undergraduates in mathematics as well as a reference for number theorists. The text here is clear and covers all the important topics like The Series of Primes, Farey Series and a Theorem of Minkowski, Irrational Numbers, Congruences and Residues, General Properties of Congruences, Congruences to Composite Moduli, The Representation of Numbers by Decimals, Continued Fractions, Fermat’s Theorem and its Consequences, Approximation of Irrationals by Rationals, The Fundamental Theorem of Arithmetic in k(l), k(i), and k(p), Some Diophantine Equations, Quadratic Fields, The Arithmetical Functions ø(n), µ(n), *d(n), *s(n), r(n), Generating Functions of Arithmetical Functions, The Order of Magnitude of Arithmetical Functions, Partitions, Kronecker’s Theorem, Geometry of Numbers, Elliptic Curves, etc.

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