Number Theory IV

1. Front Matter

   Pages i-9

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2. Introduction

   • A. N. Parshin, I. R. Shafarevich

   Pages 11-21

3. Approximation of Algebraic Numbers

   • A. N. Parshin, I. R. Shafarevich

   Pages 22-77

4. Effective Constructions in Transcendental Number Theory

   • A. N. Parshin, I. R. Shafarevich

   Pages 78-145

5. Hilbert’s Seventh Problem

   • A. N. Parshin, I. R. Shafarevich

   Pages 146-178

6. Multidimensional Generalization of Hilbert’s Seventh Problem

   • A. N. Parshin, I. R. Shafarevich

   Pages 179-208

7. Values of Analytic Functions That Satisfy Linear Differential Equations

   • A. N. Parshin, I. R. Shafarevich

   Pages 209-258

8. Algebraic Independence of the Values of Analytic Functions That Have an Addition Law

   • A. N. Parshin, I. R. Shafarevich

   Pages 259-308

9. Back Matter

   Pages 309-347

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This book was written over a period of more than six years. Several months after we finished our work, N.1. Fel'dman (the senior author of the book) died. All additions and corrections entered after his death were made by his coauthor. The assistance of many of our colleagues was invaluable during the writing of the book. They examined parts of the manuscript and suggested many improvements, made useful comments and corrected many errors. I much have pleasure in acknowledging our great indebtedness to them. Special thanks are due to A. B. Shidlovskii, V. G. Chirskii, A.1. Galochkin and O. N. Vasilenko. I would like to express my gratitude to D. Bertrand and J. Wolfart for their help in the final stages of this book. Finally, I wish to thank Neal Koblitz for having translated this text into English. August 1997 Yu. V.Nesterenko Transcendental Numbers N.1. Fel'dman and Yu. V. Nesterenko Translated from the Russian by Neal Koblitz Contents Notation ...................................................... 9 Introduction ................................................... 11 0.1 Preliminary Remarks .................................. 11 0.2 Irrationality of J2 ..................................... 11 0.3 The Number 1C' •••••••••••••••••••••••••••••••••••••••• 13 0.4 Transcendental Numbers ............................... 14 0.5 Approximation of Algebraic Numbers .................... 15 0.6 Transcendence Questions and Other Branches of Number Theory ..................................... 16 0.7 The Basic Problems Studied in Transcendental Number Theory ....................................... 17 0.8 Different Ways of Giving the Numbers ................... 19 0.9 Methods .......................... . . . . . . . . . . . . . . 20 . . . . .

• DEX
• Slate
• Transcendental numbers
• Transzendentale Zahlen
• analytic function
• approximation
• construction
• differential equation
• equation
• function
• functions
• number theory
• proof
• special function
• techniques

• Steklov Mathematical Institute, Moscow, Russia

  A. N. Parshin, I. R. Shafarevich

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