The subject of this book is also known as the circle method or the Hardy-Littlewood circle method; it was originally developed by Hardy and Ramanujan in a 1918 paper on partitions. The general idea is to write a function, whose asymptotic value is desired, in terms of a contour integral of its generating function around a circle just inside the unit circle, but then divide up the path of integration into many small arcs of different lengths and estimate each the integral along each arc separately. The arcs are classified as major or minor according to whether their contribution to the whole is expected to be large or small (not according to their length). Vinogradov’s method of trigonometrical sums is a variant of this method and is also covered in the present book.
The method has been successful on a number of problems, including the partition function, Waring’s problem, and the Goldbach conjecture. The general approach is to think of the problem as a diophantine equation and develop estimates for the number of solutions (for example, in Waring’s problem we investigate the number of ways to represent a number as the sum of a fixed number of nth powers). If we can get a positive lower bound for the number of representations, we know there is always a representation.
The present book presents various applications and refinements of the circle method. Early on it proves the classic results on Waring’s problem and the ternary Goldbach problem (every sufficiently large odd number is the sum of three primes). The bulk of the book is devoted to refining estimates for the number of solutions, and in Waring’s problem to estimate the number of nth powers required. There are also treatments of a theorem of Roth on arithmetic progressions and a theorem of Birch on solutions to homogeneous diophantine equations, and some work on diophantine approximations.
The present book concentrates on the circle method itself. Waring’s problem and the Goldbach conjecture have been the subject of much research, and there are many other approaches to these problems and much simplification of the original proofs has occurred, with expositions in a number of books. I mention here in particular Nathanson’s Additive Number Theory: The Classical Bases for both problems, including a detailed working-out of the circle method, and Davenport’s Multiplicative Number Theory for a streamlined Goldbach ternary problem. Newman’s little book Analytic Number Theory also covers many of the same problems as the present book, but with extremely short and slick proofs of the basic results (but without the finer estimates that the present book gives).
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.