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The Method of Trigonometrical Sums in the Theory of Numbers directed education is necessary toSince 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