This set of notes has been organized in such a way to create a single volume suitable for a brief introduction to one of the major developments of twentieth century mathematics, the famous Hardy-Littlewood method for studying additive problems in analytic number theory. We discuss sums of powers of integers, sums of primes, diophantine inequalities and conclude with a description of Roth's celebrated theorem on arithmetic progressions. The material represents a somewhat expanded version of selected chapters of the excellent monograph of RC Vaughan on the subject.
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