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 Pi 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 Apéry’s surprising proof of the irrationality of ζ(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 and references.
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目录
Notation Introduction 0.1 Preliminary Remarks 0.2 Irrationality of 2 0.3 The Number π 0.4 Transcendental Numbers 0.5 Approximation of Algebraic Numbers 0.6 Transcendence Questions and Other Branches of Number Theory 0.7 The Basic Problems Studied in Transcendental Number Theory 0.9 Different Ways of Giving the Numbers 0.9 Methods Chapter 1 Approximation of Algebraic Numbers §1. Preliminaries 1.1. Parameters for Algebraic Numbers and Polynomials 1.2. Statement of the Problem 1.3. Approximation of Rational Numbers 1.4. Continued Fractions 1.5. Quadratic Irrationalities 1.6. Liouville's Theorem and Liouville Numbers 1.7. Generalization of Liouville's Theorem §2. Approximations of Algebraic Numbers and Thue's Equation 2.1. Thue's Equation 2.2. The Casen=2 2.3. The Case n>3 §3. Strengthening Liouville's Theorem. First Version of Thue's Method 3.1. A Way to Bound qθ-p 3.2. Construction of Rational Approximations for a/b 3.3. Thue's First Result 3.4. Effectiveness 3.5. Effective Analogues of Theorem 1.6 3.6. The First Effective Inequalities of Baker 3.7. Effective Bounds on Linear Forms in Algebraic Numbers §4. Stronger and More General Versions of Liouville's Theorem and Thue's Theorem 4.1. The Dirichlet Pigeonhole Principle 4.2. Thue's Method in the General Case 4.3. Thue's Theorem on Approximation of Algebraic Numbers 4.4. The Non-effectiveness of Thue's Theorems §5. Further Development of Thue's Method 5.1. Siegel's Theorem 5.2. The Theorems of Dyson and Gel'fond 5.3. Dyson's Lemma 5.4. Bombieri's Theorem §6. Multidimensional Variants of the Thue-Siege1 Method 6.1. Preliminary Remarks 6.2. Siegel's Theorem 6.3. The Theorems of Schneider and Mahler §7. Roth's Theorem 7.1. Statement of the Theorem 7.2. The Index of a Polynomial 7.3. Outline of the Proof of Roth's Theorem 7.4. Approximation of Algebraic Numbers by Algebraic Numbers 7.5. The Number Ic in Itoth's Theorem 7.6. Approximation by Numbers of a Special Type 7.7. Transcendence of Certain Numbers 7.8. The Number of Solutions to the Inequality (62) and Certain Diophantine Equation §8. Linear Forms in Algebraic Numbers and Schmidt's Theorem 8.1. Elementary Estimates 8.2. Schmidt's Theorem 8.3. Minkowski's Theorem on Linear Forms 8.4. Schmidt's Subspace Theorem 8.5. Some Facts from the Geometry of Numbers §9. Diophantine Equations with the Norm Form 9.1. Preliminary Remarks 9.2. Schmidt's Theorem §10. Bounds for Approximations of Algebraic Numbers in Non-archimedean Metrics 10.1. Mahler's Theorem 10.2. The Thue-Mahler Equation 10.3. Further Non-effective Results Chapter 2 Effective Constructions in Transcendental Number Theory §1. Preliminary Remarks 1.1. Irrationality of e 1.2. Liouville's Theorem 1.3. Hermite's Method of Proving Linear Independence of a Set of Numbers 1.4. Siegel's Generalization of Hermite's Argument 1.5. Gel'fond's Method of Proving That Numbers Are Transcendental §2. Hermite's Method 2.1. Hermite's Identity 2.2. Choice of f(x) and End of the Proof That e is Transcendental 2.3. The Lindemam and Lindemann-Weierstrass Theorems 2.4. Elimination of the Exponents 2.5. End of the Proof of the Lindemann-Weierstrass Theorem 2.6. Generalization of Hermite's Identity §3. Functional Approximations 3.1. Hermite's Functional Approximation for ez 3.2. Continued Fraction for the Gauss Hypergeometric Function and Pad6 Approximations 3.3. The HermitePad6 Functional Approximations §4. Applications of Hermite's Simultaneous Functional Approximations 4.1. Estimates of the Transcendence Measure of e 4.2. Transcendence of eπ 4.3. Quantitative Refinement of the Lindemann-Weierstrass Theorem 4.4. Bounds for the Transcendence Measure of the Logarithm of an Algebraic Number 4.5. Bounds for the Irrationality Measure of n and Other Numbers 4.6. Approximations to Algebraic Numbers §5. Bounds for Rational Approximations of the Values of the Gauss Hypergeometric Function and Related Functions 5.1. Continued Fractions and the Values of ex 5.2. Irrationality of π 5.3. Maier's Results 5.4. Further Applications of Pad6 Approximation 5.5. Refinement of the Integrals 5.6. Irrationality of the Values of the Zeta-Function and Bounds on the Irrationality Exponent §6. Generalized Hypergeometric Functions 6.1. Generalized Hermite Identities 6.2. Unimprovable Estimates 6.3. Ivankov's Construction §7. Generalized Hypergeometric Series with Finite Radius of Convergence 7.1. Functional Approximations of the First Kind 7.2. Functional Approximations of the Second Kind §8. Remarks Chapter 3 Hilbert's Seventh Proble §1. The Euler-Hilbert Problem 1.1. Remarks by Leibniz and Euler 1.2. Hilbert's Report §2. Solution of Hilbert's Seventh Problem 2.1. Statement of the Theorems 2.2. Gel'fond's Solution 2.3. Schneider's Solution 2.4. The Real Case 2.5. Laurent's Method §3. Transcendence of Numbers Connected with Weierstrass hnctions 3.1. Preliminary Remarks 3.2. Schneider's Theorems 3.3. Outline of Proof of Schneider's Theorems §4. General Theorems 4.1. Schneider's General Theorems 4.2. Consequences of Theorem 3.17 4.3. Lang's Theorem 4.4. Schneider's Work and Later Results on Abelian Functions §5. Bounds for Linear Forms with Two Logarithms 5.1. First Estimates for the Transcendence Measure of ab and lnα/lnβ 5.2. Refinement of the Inequalities (19) and (20) Using Gel'fond's Second Method 5.3. Bounds for Transcendence Measures 5.4. Linear Forms with Two Logarithms 5.5. Generalizations to Non-archimedean Metrics 5.6. Applications of Bounds on Linear Forms in Two Logarithms §6. Generalization of Hilbert's Seventh Problem to Liouville Numbers 6.1. Ricci's Theorem 6.2. Later Results §7. Transcendence Measure of Some Other Numbers Connected with the Exponential Function 7.1. Logarithms of Algebraic Numbers 7.2. Approximation of Roots of Certain Transcendental Equations §8. Transcendence Measure of Numbers Connected with Elliptic Functions 8.1. The Case of Algebraic Invariants 8.2. The Case of Algebraic Periods 8.3. Values of p(z) at Non-algebraic Points Chapter 4 Multidimensional Generalization of Hilbert's Seventh Problem §1. Linear Forms in the Logarithms of Algebraic Numbers 1.1. Preliminary Remarks 1.2. The First Effective Theorems in the General Case 1.3. Baker'sMethod 1.4. Estimates for the Constant in (8) 1.5. Methods of Proving Bounds for Λ，Λo，and Λ1 1.6. A Special Form for the Inequality 1.7. Non-archimedean Metrics §2. Applications of Bounds on Linear Forms 2.1. Preliminary Remarks 2.2. Effectivization of Thue's Theorem 2.3. Effective Strengthening of Liouville's Theorem 2.4. The Thue-Mahler Equation 2.5. Solutions in Special Sets 2.6. Catalan's Equation 2.7. Some Results Connected with Fermat's Last Theorem 2.8. Some Other Diophantine Equations 2.9. The abc-Conjecture 2.10. The Class Number of Imaginary Quadratic Fields 2.11. Applications in Algebraic Number Theory 2.12. Recursive Sequences 2.13. Prime Divisors of Successive Natural Numbers 2.14. Dirichlet Series §3. Elliptic Functions 3.1. The Theorems of Baker and Coates 3.2. Masser's Theorems 3.3. Further Results 3.4. Wiistholz's Theorems §4. Generalizations of the Theorems in §1 to Liouville Numbers 4.1. Walliser's Theorems 4.2. Wiistholz's Theorems Chapter 5 Values of Analytic Functions That Satisfy Linear Differential Equations §1. E-Functions 1.1. Siegel's Results 1.2. Definition of E-Functions and Hypergeometric E-Functions 1.3. Siegel's General Theorem 1.4. Shidlovskii's Fundamental Theorem §2. The Siegel-Shidlovskii Method 2.1. A Technique for Proving Linear and Algebraic Independence 2.2. Construction of a Complete Set of Linear Forms 2.3. Nonvanishing of the Functional Determinant 2.4. Concluding Remarks §3. Algebraic Independence of the Values of Hypergeometric E-Functions 3.1. The Values of EFunctions That Satisfy First，Second，and Third Order Differential Equations 3.2. The Values of Solutions of Differential Equations of Arbitrary Order §4. The Values of Algebraically Dependent E-Functions 4.1. Theorem on Equality of Transcendence Degree 4.2. Exceptional Points §5. Bounds for Linear Forms and Polynomials in the Values of EFunctions 5.1. Bounds for Linear Forms in the Values of E-Functions 5.2. Bounds for the Algebraic Independence Measure §6. Bounds for Linear Forms that Depend on Each Coefficient 6.1. A Modification of Siegel's Scheme 6.2. Baker's Theorem and Other Concrete Results 6.3. Results of a General Nature §7. G-Functions and Their Values 7.1. G-Functions 7.2. Canceling Factorials 7.3. Arithmetic Type 7.4. Global Relations 7.5. Chudnovsky's Results Chapter 6. Algebraic Independence of the Values of Analytic Functions That Have an Addition Law §1. Gel'fond's Method and Results 1.1. Gel'fond's Theorems 1.2. Bound for the Transcendence Measure 1.3. Gel'fond's "Algebraic Independence Criterion" and the Plan of Proof of Theorem 6.3 1.4. Further Development of Gel'fond's Method 1.5. Fields of Finite Transcendence Type §2. Successive Elimination of Variables 2.1. Small Bounds on the aanscendence Degree 2.2. An Inductive Procedure §3. Applications of General Elimination Theory 3.1. Definitions and Basic Facts 3.2. Philippon's Criterion 3.3. Direct Estimates for Ideals 3.4. Effective Klbert Nullstellensatz §4. Algebraic Independence of the Values of Elliptic Functions 4.1. Small Bounds for the l'kanscendence Degree 4.2. Elliptic Analogues of the Lindemann-Weierstrass Theorem 4.3. Elliptic Generalizations of Hilbert's Seventh Problem §5. Quantitative Results 5.1. Bounds on the Algebraic Independence Measure 5.2. Bounds on Ideals，and the Algebraic Independence Measure 5.3. The Approximation Measure Bibliography Index
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