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Selected Works: Ivan Matveevič Vinogradov

Ivan Matveevič Vinogradov
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Springer Collected Works in Mathematics
[Reviewed by
Allen Stenger
, on

I. M. Vinogradov (1891–1983) was a Russian number theorist. His early work dealt with the distribution of power residues and primitive roots, but in the 1930s he developed his method of trigonometric sums; most of his life work consisted of different applications of this method. His best known result is probably a solution in 1937 to the Goldbach ternary problem, proving that every sufficiently large odd number is the sum of three primes. He was also the author of an introductory text, Elements of Number Theory.

The present volume contains sixteen papers, spanning his career, that he selected as his most important, and the complete texts of two monographs on his method. All works originally appeared in Russian and are given here in new English translations. This is a 2014 reprint of the 1985 Springer English-language edition.

Vinogradov’s 1947 monograph The Method of Trigonometric Sums in Number Theory is best known through the 1954 English-language version, which is still in print from Dover. Vinogradov later completely rewrote it and split it into two volumes, a 1980 volume with the same title that covered the use of Vinogradov’s mean-value theorem (a result about Weyl sums), and a 1976 volume, titled Special Variants of the Method of Trigonometric Sums, that dealt with other problems solved through his methods without using the mean-value theorem. Both monographs are included here in English translation. Neither monograph is very systematic, and they are focused on specific problems rather than general techniques.

The book concludes with an 8-page overview of Vinogradov’s life and work by K. K. Mardzhanishvili and a complete bibliography of his works.

Many advances have been made on these problems since Vinogradov did his work, and there are much better places to learn about them than his original papers and monographs. I mention in particular Vaughan’s The Hardy-Littlewood Method, which is very systematic, shows Vinogradov’s method in the context of the circle method, and proves all the best-known results.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

A New Method of Deriving Asymptotic Expressions for Arithmetic Functions.- On the Mean Value of the Number of Classes of Proper Primitive Forms of Negative Discriminant.- On the Distribution of Power Residues and Non-Residues.- An Elementary Proof of a General Theorem in Analytic Number Theory.- On the Distribution of Indices.- On a Bound for the Least nth Power Non-Residue.- On the Distribution of Fractional Parts of Values of a Function of One Variable.- On the Distribution of Fractional Parts of Values of a Function of Two Variables.- On Waring's Theorem.- A New Estimate for G (n) in Waring's Problem.- On the Upper Bound for G (n) in Waring's Problem.- New Estimates for Weyl Sums.- Representation of an Odd Number as the Sum of Three Primes.- Estimates of Certain Simple Trigonometric Sums with Prime Numbers.- Certain Problems in Analytic Number Theory.- On the Distribution of the Fractional Parts of Values of a Polynomial.- The Method of Trigonometric Sums in Number Theory.- Special Variants of the Method of Trigonometric Sums.- Ivan Matveevic Vinogradov: A Brief Outline of His Life and Works.- A Chronological List of I. M. Vinogradov's Works.