By Yuan Wang

The circle process has its genesis in a paper of Hardy and Ramanujan (see [Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrep resenting numbers as sums ofsquares. Later, in a chain of papers starting in 1920entitled "some difficulties of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and constructed systematically a brand new analytic procedure, the circle approach in additive quantity thought. the main well-known difficulties in advert ditive quantity conception, particularly Waring's challenge and Goldbach's challenge, are handled of their papers. The circle process can be referred to as the Hardy-Littlewood strategy. Waring's challenge should be defined as follows: for each integer ok 2 2, there's a quantity s= s( okay) such that each optimistic integer N is representable as (1) the place Xi arenon-negative integers. This statement wasfirst proved by way of Hilbert [1] in 1909. utilizing their strong circle procedure, Hardy and Littlewood received a deeper consequence on Waring's challenge. They demonstrated an asymptotic formulation for rs(N), the variety of representations of N within the shape (1), particularly ok 1 only if eight 2 (k - 2)2 - +5. the following

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We have V2 ~ Ton- l (To + N( a)l/n - To + N( a)l/n) ~ Ton- l N( a)l/n Denote by V; the volumes of (V;) (i VI and Vo = 1 dx = (Vo) II Ti IIm JU2+V2~T;' f dudv = 7I"r i 2 IT i=l Ti , where dx = TIl~i~n dXi. 2 N(a)v'l5 The lemma follows. 15) is replaced by 1:::; i :::; n. 3. 15) where 0 < v(l) < C6N(V)1/n, Iv(m)1 C6 = c6(K) N(v):::; Tn, and I" is a given number in a residue class mod a. Then N ( 1 where To < cr,N(v)l/n, a, Tn) n 1 ) c7(K) Tn 0 ( To = N( a) + N( a)l-l/n ' = max (N(a)l/n, T). 2 the condition v E a can be changed to v == I" (mod a).

An of a such that 1 ~ i,j ~ n. Let A be an ideal class containing a. Then a is a product of a fixed ideal ao E A, and a number a of K: a = aao. Let AI, ... ,An be a basis of ao. Then ai = aAi (1 ~ i ~ n) is a basis of a. Set 1 ~ i ~ n. 1 that there exists a totally nonnegative unit 7] such that 1 ~ i ~ n. Put a' = a7]. Then 1 i ~ ~ n. We may use a' instead of a. Hence we may suppose that 1~ i Therefore ~ n. laji) I = la(i),Xji)I ~ IN(aW/ n ~ N(a)l/n. 2) For any x = (Xl,"" x n ), where Xl are real numbers and X m = xm +r2 ' we define Xl(X) = Xl, Xm(x) = Re(x m ), X m+r2 (x) = Im(x m), and X(x) = (Xl(x), ...

2). This assertion is trivially true whenever la(i) I ~ D-l/Z, (1 ~ i ~ n). 6) for some index j + rz, and we must prove that ~ rl Since q-l is integral, we have N( q) ~ 1. 8) in the case j > rl. 6). 5) if the pair a, ~ were in 6. 4) are not all satisfied. 8), 0< IIKall ~ h, and so we have IK(i)lla(i)~(i) h la(j)~U) ,8(i) I< i, i =I j or i =I j,j + rz, - ,8U)1 ~ IKU)I- 1 ~ D- 1 / Z • The lemma is proved. 1. 9) = rl + rz. Let 11, ... " r be a set of real numbers satisfying L Ii + 2 L 1m = O. pter 3.