The method of trigonometrical sums in the theory of numbers. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport. Reprint of the 1954 translation. (English) Zbl 1093.11001

Vinogradov, I. M.

Mineola, NY: Dover Publications (ISBN 0-486-43878-3/pbk). x, 180 p. (2004).

This is an unabridged republication of the 1954 edition (see Zbl 0055.27504). The translators write: “Since 1934 the analytic theory of numbers has been largely transformed by the work of Vinogradov. This work, which has led to remarkable new results, is characterized by its supreme ingenuity and great power.”
Contents: 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 \(G(n)\) in 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 \(\alpha p\).

Reviewer: Florin Nicolae (Berlin)

MSC:

11-02	Research exposition (monographs, survey articles) pertaining to number theory
11L03	Trigonometric and exponential sums (general theory)
11L40	Estimates on character sums

Keywords:

trigonometric sums; Waring’s problem; Goldbach’s problem

Citations:

Zbl 0055.27504