Ebook: Number Theory IV: Transcendental Numbers

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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This book is a survey of the most important directions of research in transcendental number theory - the theory of irrationality, transcendence, and algebraic independence of various numbers. There is a special emphasis on the transcendence properties of the values of special functions. The book contains few complete proofs, but rather gives conceptual discussions of the central ideas behind the proofs. For a reader who has no specific background in transcendental number theory the book provides both an overview of the basic concepts and techniques and also a guide to the most important results and references.