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 sequence of papers starting in 1920entitled "some difficulties of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and built systematically a brand new analytic procedure, the circle approach in additive quantity thought. the main recognized difficulties in advert ditive quantity conception, particularly Waring's challenge and Goldbach's challenge, are taken care of of their papers. The circle procedure is usually known as the Hardy-Littlewood approach. Waring's challenge should be defined as follows: for each integer okay 2 2, there's a quantity s= s( okay) such that each confident integer N is representable as (1) the place Xi arenon-negative integers. This statement wasfirst proved through Hilbert [1] in 1909. utilizing their strong circle process, Hardy and Littlewood received a deeper end result on Waring's challenge. They confirmed 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. right here

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L :L ... :L 2k 1 P. Pl Al(P) :L ... (p) where and M is equal to the number of solutions of the system of congruences 1 ~ h ~ k. k) (mod p), since pI k!. Then we have that AI, ... 1, . k and thus Consequently we have S(f(A) I ,p ) 12k < k· k! ( )2k - N(p)(N(p)_l)N p ~ and the lemma is proved. 2k . k! 7. 1) Proof. Let b = (kak, ... ,al)' Then alb. Suppose that ptllba- 1 . \ runs over a complete residue system (mod p). 2. 3) IT t ~ 1, then pt divides at least one of the integers k, . 4) Suppose that e < 2(t + 1).

27) into two parts: where E 1 is the sum taken over all t with tj = 0 for all j E J 1 + J2 , and E 2 consists of the other terms. Since J3 and J1 + J2 are not empty we have E 1 ~ ATon - 1 ~ AB n - 1. 28). 29) 2) Estimation of 52. L) E 42 Chapter 3. Weyl's Sum 82 L ~ {9, }~{o} First we assume that Wo Since la(t) I W(g)min--,· tEJa Tt Igt > 1. Then Igil ~ Gi, we have . la(t) 1 " , , ' ~ la(g) I mm-- ~ LJ LJ LJ - - , a {9,}~{O} tEJ Ttlgtl gOa (g) 9,=1 Tgiggi " LJ where ~(g). means a sum taken over gt (t E J3 , t -::f q).

2) If log 11](1)1 have < 0, then log 11](1)1 :$ -~h(1]). 5 A Lemma on the set D( x) 3) If log 17P)1 < 0, then min log 17](i) 1 = log 17](1)\ ~ -~h(7]). 4) If log 17](1)1 ~ 0, then log 17](1)1 ~ ~h(7]). It follows from N(7]) 51 = 1 that o ~ (n -l)minlog 17](i) \ + log 17](1)1 ~ (n -l)minlog 17](i) 1 + ~h(7]). Set C4 = 2C6/ ( ,jii(n - 1)). The lemma follows. 0 Let Xi = ye x :, 1 ~ i ~ n, where y > O. Then H' becomes H(y), and C~(t) becomes C'1(y, t). 6. 7. Suppose that X > O. Then D(x) C UC'1(C3XV;I), '1 where v'1 = max 17](i)l.