The Method of Trigonometrical Sums in the Theory of Numbers

I. M. Vinogradov

• Publisher: Dover Publications
• Year: 2013
• Print ISBN:
  9780486438788
• eBook ISBN:
  9780486154527
• Topic: Mathematik
• Subtopic: Trigonometrie

Since the 1930s, the analytic theory of numbers has been transformed by the influence of I. M. Vinogradov, and this text for upper-level undergraduates and graduate students testifies to its author's ingenuity and to the effectiveness of his methods. Starting with a discussion of general lemmas, it advances to an investigation of Waring's problem, including explorations of singular series, the contribution of the basic intervals, and an estimate for G(n). Further topics include approximation by the fractional parts of the values of a polynomial, estimates for Weyl sums, the asymptotic formula in Waring's problem, the distribution of the fractional parts of the values of a polynomial, estimates for the simplest trigonometrical sums with primes, and Goldbach's problem. 1954 edition.

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface by the Translators
• Contents
• Notation
• Introduction
• Note on Vinogradov’s Method
• I. General Lemmas
• II. The Investigation of the Singular Series in Waring’s Problem
• III. The Contribution of the Basic Intervals in Waring’s Problem
• IV. An Estimate for G(n) Waring’s Problem
• V. Approximation by the Fractional Parts of the Values of a Polynomial
• VI. Estimates for Weyl Sums
• VII. The Asymptotic Formula in Waring’s Problem
• VIII. The Distribution of the Fractional Parts of the Values of a Polynomial
• IX. Estimates for the Simplest Trigonometrical Sums with Primes
• X. Goldbach’s Problem
• XI. The Distribution of the Fractional Parts of the Values of the Function αp