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The Method of Trigonometrical Sums in the Theory of Numbers

I. M. Vinogradov
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Sidney Graham
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The Russian mathematician I.M. Vinogradov was one of the giants of analytic number theory in the twentieth century, and his method of trigonometrical sums is his legacy. Vinogradov wrote two expository accounts of his method; the second was published in 1947. Interscience published an English translation of this in 1954; the translation was done by K.F. Roth and A.M. Davenport. Dover has now published this inexpensive paperback edition.

There are actually two distinct methods associated with Vinogradov. The first is his method for estimating the sum of f(p), where f is some function and p runs over primes. The first big success for this technique was Vinogradov's 1937 proof of the ternary Goldbach problem; i.e., that every sufficiently large odd number is a sum of three primes. The second is his mean value theorem, which lead to improved estimates in Waring's problem and improved zero-free regions for the Riemann zeta-function.

Vinogradov was a mathematician with great insight and technical power, but his exposition can be very convoluted, unmotivated, and difficult to follow. The novice student in analytic number theory would be better served by first reading some of the more recent treatments of these subjects. A good alternative would be The Hardy-Littlewood Method by R.C. Vaughan, which covers both aspects of Vinogradov's method and includes Vaughan's simplified verions of the method for estimating sums over primes. Another good alternative is Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, by H.L. Montgomery; Chapter 4 begins with a result of Mordell that motivated Vinogradov's mean value theorem. On the other hand, this book will be quiet useful to those who are tracing the history of the subject or who want to see the seminal ideas of Vinogradov in their original form.

S. W. Graham is currently Professor of Mathematics at Central Michigan University. He can be reached at [email protected].

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