Table of contents

CONTENTS Introduction § 1. The approximation of algebraic numbers by rational and algebraic numbers § 2. The method of Hermite and Lindemann § 3. Gel'fond's method § 4. Siegel's method § 5. The further development of Siegel's method § 6. Conclusion References

CONTENTS Introduction § 1. The approximate calculation of multiple sums § 2. The properties of optimal coefficients References

CONTENTS Basic notation Introduction Chapter I. Auxiliary lemmas § 1. The functions , , , and their properties § 2. Estimates for the solutions of some approximating integro-difference equations § 3. Asymptotic behaviour of the solutions of some differential-difference equations Chapter II. An estimate for and its applications § 1. An estimate for . Examples § 2. Local distribution laws for additive arithmetic functions § 3. Integral theorems § 4. Equidistribution of the fractional parts of some additive functions Chapter III. Generalized problem of the numbers with small and large prime factors, and applications § 1. Statement of the problem and some special cases § 2. Solution of the general problem § 3. Upper bounds for sums extended over numbers with small prime factors § 4. Numbers with small prime factors § 5. Applications to the probabilistic theory of numbers Chapter IV. The case of arithmetic progressions with increasing differences § 1. Solution of the general problem for arithmetic progressions § 2. Some applications References