Number Theory IV: Transcendental Numbers

A. N. Parshin, I. R. Shafarevich (auth.), A. N. Parshin, I. R. Shafarevich (eds.)9783540614678, 3540614672

This book is a survey of the most important directions of research in transcendental number theory. The central topics in this theory include proofs of irrationality and transcendence of various numbers, especially those that arise as the values of special functions. Questions of this sort go back to ancient times. An example is the old problem of squaring the circle, which Lindemann showed to be impossible in 1882, when he proved that $Öpi$ is a transcendental number. Euler’s conjecture that the logarithm of an algebraic number to an algebraic base is transcendental was included in Hilbert’s famous list of open problems; this conjecture was proved by Gel’fond and Schneider in 1934. A more recent result was ApÖ’ery’s surprising proof of the irrationality of $Özeta(3)$ in 1979. The quantitative aspects of the theory have important applications to the study of Diophantine equations and other areas of number theory. For a reader interested in different branches of number theory, this monograph provides both an overview of the central ideas and techniques of transcendental number theory, and also a guide to the most important results.

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Table of contents :
Front Matter….Pages i-9
Introduction….Pages 11-21
Approximation of Algebraic Numbers….Pages 22-77
Effective Constructions in Transcendental Number Theory….Pages 78-145
Hilbert’s Seventh Problem….Pages 146-178
Multidimensional Generalization of Hilbert’s Seventh Problem….Pages 179-208
Values of Analytic Functions That Satisfy Linear Differential Equations….Pages 209-258
Algebraic Independence of the Values of Analytic Functions That Have an Addition Law….Pages 259-308
Back Matter….Pages 309-347