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An Introduction to the Circle Method

M. Ram Murty and Kaneenika Sinha
Publication Date: 
Number of Pages: 
Student Mathematical Library
[Reviewed by
Allen Stenger
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The circle method is a technique, originated by Ramanujan, to estimate the coefficients in a generating function by expressing the coefficient in terms of an integral of the function around the unit circle in the complex plane and estimating the integral. It has been successfully applied to many problems in additive number theory, including Waring’s Problem and the ternary Goldbach Conjecture (the two that are shown in this book), as well as estimating the number of partitions, and various problems in Diophantine equations.
The present book is aimed at upper-division undergraduates and beginning graduate students, and explains everything from scratch and writes out nearly all of the calculations. Despite the power of the method, it is not conceptually difficult, nor does its application require any advanced knowledge (beyond elementary number theory and a little complex analysis), but the
proofs are very intricate and understanding them requires close attention. I think the book does a good job of explaining the method and the two applications. The first third of the book is a development of elementary number theory and of the Prime Number Theorem, and I think most readers who are interested in the circle method will have already learned these and can skip
over this part of the book.
The Waring Problem, which states that for each positive integer power \( k \), every positive integer is the sum of a bounded number of \( k \)th powers of positive integers (a different bound for each \( k \)). For example, it has long been know that every positive integer is the sum of four squares, and the Waring Problem generalizes this to arbitrary exponents. The Waring Problem was first solved by Hilbert in 1909 using an intricate argument involving 25-fold multiple integrals. The book presents two treatments of the Waring Problem.  The first, due to Linnik, only takes a few pages and shows that the bound exists. It is not strictly speaking a circle-method proof, but it uses many of the same ingredients that applications of the circle method do. The second treatment develops an asymptotic formula for the number of representations of a positive integer as a sum of a specified number of \( k \)th powers; from this it follows that there is always a representation if we allow enough summands
The ternary Goldbach Conjecture states that every odd number greater than 5 is the sum of three primes. (The binary Goldbach Conjecture states that every even number greater than 2 is the sum of two primes; it implies the ternary conjecture.) The ternary Goldbach Conjecture was proved by I. M. Vinogradov in 1937 for all sufficiently large numbers, and in recent years
Harald Helfgott and others have extended this to prove it for all numbers.  The present book presents a proof of Vinogradov’s result. It also includes some discussion of the binary Goldbach Conjecture; the circle method does not help much on this problem, but much progress has been made using other methods, especially sieves.
There are numerous exercises scattered throughout the book (but no solutions). These are intended to increase the reader’s familiarity with the concepts being introduced and are not very difficult.  The circle method goes by a variety of names, most commonly the Hardy–Littlewood method as in the title of Vaughan’s book. They made the greatest development of the method, in a series of papers in the 1920s. Sometimes the naming includes Ramanujan, who introduced the method in a letter to Hardy, and who was the co-author with Hardy of the first published
paper using the method (to get an asymptotic formula for the number of partitions), or includes I. M. Vinogradov, who simplified and extended the method. Vinogradov truncated the generating function to get a finite sum and parameterized it so that it is a trigonometric polynomial rather than a function of a complex variable. His method is often called the method of
trigonometric sums rather than the circle method.
The definitive work on the circle method is Vaughan’s The Hardy–Littlewood Method. This covers everything in the present book, goes into greater depth, and shows how to apply the circle method to a variety of other problems. A very valuable book is Nathanson’s Additive Number Theory: The Classical Bases. Like the present work, it develops everything from scratch and writes out all the details, but it is focussed specifically on the Waring and Goldbach problems and brings to bear many methods in addition to the the circle method.
Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal web site is His mathematical interests are number theory and classical analysis.