ANALYTIC METHODS IN NUMBER THEORY

There is no surprise that arithmetic properties of integral ('whole') numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other 'better known' numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.

The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of π and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.

Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible — almost popular — level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.

Contents:

• Preface
• Numbers and q-Numbers
• Prime Number Theorem
• Riemann's Zeta Function and Its Multiple Generalisation
• Continued Fractions
• Dirichlet's Theorem on Primes in Arithmetic Progressions
• Algebraic and Transcendental Numbers. The Transcendence of e and π
• Irrationality of Zeta Values
• Hilbert's Seventh Problem
• Schinzel–Zassenhaus Conjecture
• Creative Microscoping
• Bibliography
• Index

Readership: Graduates, researchers and enthusiasts in number theory, complex analysis, and special functions; suitable for teaching graduate courses in number theory and self-study.

Key Features:

• A review of different analytic techniques used in number theory
• A unique collection of topics including classical ones (distribution of primes, primes in arithmetic progressions, continued fractions, transcendence of e, π and resolution of Hilbert's seventh problem) and recent ones (irrationality questions for the values of Riemann's zeta function at odd positive integers, Dimitrov's resolution of the Schinzel — Zassenhaus ex-conjecture about size of non-cyclotomic algebraic integers and analytic methods for proving congruences for truncated hypergeometric sums), which are hardly covered in other texts
• A supply of the material with varying difficulty-level problems
• A book by a world expert in the theory of irrational and transcendental numbers
• A material suitable for both teaching graduate courses in number theory and self-study