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The circle method has its genesis in a paper of Hardy and Ramanujan (see [Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrep resenting numbers as sums ofsquares. Later, in a series of papers beginning in 1920entitled "some problems of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and developed systematically a new analytic method, the circle method in additive number theory. The most famous problems in ad ditive number theory, namely Waring's problem and Goldbach's problem, are treated in their papers. The circle method is also called the Hardy-Littlewood method. Waring's problem may be described as follows: For every integer k 2 2, there is a number s= s( k) such that every positive integer N is representable as (1) where Xi arenon-negative integers. This assertion wasfirst proved by Hilbert [1] in 1909. Using their powerful circle method, Hardy and Littlewood obtained a deeper result on Waring's problem. They established an asymptotic formula for rs(N), the number of representations of N in the form (1), namely k 1 provided that 8 2 (k - 2)2 - +5. Here
List of contents
1. The Circle Method and Waring's Problem.- 1.1 Introduction.- 1.2 Farey Division.- 1.3 Auxiliary Lemmas.- 1.4 Major Arcs.- 1.5 Singular Integral.- 1.6 Singular Series.- 1.7 Proof of Lemma 1.12.- 1.8 Proof of Theorem 1.1.- Notes.- 2. Complete Exponential Sums.- 2.1 Introduction.- 2.2 Several Lemmas.- 2.3 Mordell's Lemma.- 2.4 Fundamental Lemma.- 2.5 Proof of Theorem 2.1.- 2.6 Proof of Theorem 2.2.- Notes.- 3. Weyl's Sums.- 3.1 Introduction.- 3.2 Proof of Theorem 3.1.- 3.3 A Lemma on Units.- 3.4 The Asymptotic Formula for N(a,T).- 3.5 A Sum.- 3.6 Mitsui's Lemma.- 3.7 Proof of Theorem 3.3.- 3.8 Proof of Lemma 3.6.- 3.9 Continuation.- Notes.- 4. Mean Value Theorems.- 4.1 Introduction.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 4.4 A Lemma on the Set D.- 4.5 A Lemma on the Set D(x).- 4.6 Fundamental Lemma.- 4.7 Proof of Lemma 4.1.- Notes.- 5. The Circle Method in Algebraic Number Fields.- 5.1 Introduction.- 5.2 Lemmas.- 5.3 Asympotic Expansion forSi (?, T).- 5.4 Further Estimates on Basic Domains.- 5.5 Proof of Theorem 5.1.- 5.6 Proof of Theorem 5.2.- Notes.- 6. Singular Series and Singular Integrals.- 6.1 Introduction.- 6.2 Product Form for Singular Series.- 6.3 Singular Series and Congruences.- 6.4 p-adic Valuations.- 6.5 k-th Power Residues.- 6.6 Proof of Theorem 6.1.- 6.7 Monotonic Functions.- 6.8 Proof of Theorem 6.2.- Notes.- 7. Waring's Problem.- 7.1 Introduction.- 7.2 The Ring Jk.- 7.3 Proofs of Theorems 7.1 and 7.2.- 7.4 Proof of Theorem 7.3.- 7.5 Proof of Theorem 7.4.- Notes.- 8. Additive Equations.- 8.1 Introduction.- 8.2 Reductions.- 8.3 Contraction.- 8.4 Derived Variables.- 8.5 Proof of Theorem 8.1.- 8.6 Proof of Theorem 8.2.- 8.7 Bounds for Solutions.- Notes.- 9. Small Nonnegative Solutions of Additive Equations.- 9.1 Introduction.- 9.2Hurwitz's Lemma.- 9.3 Reductions.- 9.4 Continuation.- 9.5 Farey Division.- 9.6 Supplementary Domain.- 9.7 Basic Domains.- 9.8 Proof of Theorem 9.1.- Notes.- 10. Small Solutions of Additive Equations.- 10.1 Introduction.- 10.2 Reductions.- 10.3 Continuation.- 10.4 Farey Division.- 10.5 Supplementary Domain.- 10.6 Basic Domains.- 10.7 Proof of Theorem 10.1.- Notes.- 11. Diophantine Inequalities for Forms.- 11.1 Introduction.- 11.2 A Single Additive Form.- 11.3 A Variant Circle Method.- 11.4 Continuation.- 11.5 Proof of Lemma 11.1.- 11.6 Linear Forms.- 11.7 A Single Form.- 11.8 Proof of Theorem 11.1.- Notes.- References I.- References II.
