By Chaohua Jia, Kohji Matsumoto

Contains a number of survey articles on best numbers, divisor difficulties, and Diophantine equations, in addition to examine papers on numerous elements of analytic quantity conception difficulties.

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2 satisfying log D << log X , k be a jixed natural number, t be a fixed non-negative real number, and let r = r(d) and b = b(d) be integers with r > 0 and (r, b) = 1 for each natural number d I D. Also suppose that q and a are coprime integers satisfying lqa - a1 $ x - ~ / and ~ 1 5 q 5 x k I 2 . Then one has X a r ( r ) t (r +),(X k 1rdka - bl) d< D -i<< 2 T ( ~ ) ~ + 'log x X (9 xk1qa - al)l/k + + Proof. Let V be the set of all the natural numbers d 5 D such r 5 x k I 2 / ( 3 D k ) and lrdka - b1 < 1/(3xkI2).

Kaplansky, Sums of squares, cubes and higher powers. Experiment. Math. 4 (l995), 169-173. [22] K. Kawada, Note on the sum of cubes of primes and an almost prime. Arch. Math. 69 (1997), 13-19. [23] K. Prachar , ~ b e ein r Problem vom Waring-Goldbach 'schen Typ 11. Monatsh. Math. 57 (1953) 113-116. 91 [24] K. F. Roth, Proof that almost all positive integers are sums of a square, a cube and a fourth power. J . London Math. Soc. 24 (1949), 4-13. [25] K. F. Roth, A problem in additive number theory.

Pa(4 = 1, a = p1p2p3,NOLI pl 0, otherwise. < N P I p2 < p3, ( a , N ) = 1; 1, a = P ~ P ~ P ~ , NI PPl 0, otherwise. < p2 < p3, ( a , N ) = 1; 1, a = pip2p3n,NOL5 pl 0, otherwise. < p2 < p3 < N P ,(a,N ~ ; ' P ( ~=~ 1;) ) 104 I ANALYTIC NUMBER THEORY A On Chen's theorem P 105 Proof. Since the second inequlity can be deduced from the first one easily, so it suffice to prove the first inequality. Let Then On the other hand, Combining the above arguments we complete the proof of Lemma 5. Lemma 6. For X(a) = 0 = 1 - 1 1 -5 1 ~ (a) 2 - m (a) + ,p4 (a); 106 ANALYTIC NUMBER THEORY Proof.