In this post, we will see the book* The Method Of Trigonometric Sums In The Theory Of Numbers* by I. M. Vinogradov.

# About the book

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.

Vinogradov has expounded his method and its applications in a series of papers and in two monographs, which appeared in 1937 and 1947. The present book is a translation of the second of these monographs, which incorporated the improvements effected by the author during the intervening ten years.

The text has been carefully revised and to some extent rewritten. The more difficult arguments have been set out in greater detail.

Notes have been added, in which we mention the more important changes made and comment on the subject-matter; we hope these will be of assistance or of interest to the reader.

The book was translated from Russian by K. F. Roth and Anne Davenport and was published in 1954.

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You can get the book here.

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# Contents

Preface by the Translators V

Notation ix

Introduction 1

Note on Vinogradov’s Method by the Translators 19

## I. General Lemmas 21

Notes 42

## II. The Investigation of the Singular Series in Waring’s Problem 45

Notes 54

## III. The Contribution of the Basic Intervals in Waring’s Problem 55

Notes 61

## IV. An Estimate for *G(n)* in Waring’s Problem 62

Notes 69

## V. Approximation by the Fractional Parts of the Values of a Polynomial 71

Notes 81

## VI. Estimates for Weyl Sums 82

Notes 113

## VII. The Asymptotic Formula in Waring’s Problem 117

Notes 123

## VIII. The Distribution of the Fractional Parts of the Values of a Polynomial 124

Notes 127

## IX. Estimates for the Simplest Trigonometrical Sums with Primes 128

Notes 162

## X. Goldbach’s Problem 163

Notes 175

## XI. The Distribution of the Fractional Parts of the Values of the Function 𝛼𝓅 177

Notes 180

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