By Chaohua Jia, Kohji Matsumoto

Contains numerous survey articles on top numbers, divisor difficulties, and Diophantine equations, in addition to learn papers on numerous points of analytic quantity thought difficulties.

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Since ~ ( n ) 2"("), the lemma follows. > The proofs of Theorems 1 and 2 (i). 16), we may write where By the definition, we see that w, (p) = Bp(p,n ) / Bl (p, n ) or 1, according to p Y or p > Y, and also that w,(~') = wn(p) for all 1 1. Then we may confirm that < so that &(n) = &(n; [O, 11) = Rd(n; 1)31) This time we set s = 1, k = 2, > + Rd(n;m). 2). nd that for every integer n with N 5 n 5 (6/5)N. To facilitate our subsequent description, we denote by N(5) the set of all the odd integers in the interval [N, (6/5)N], and put N(4) = Nl n > for all primes p and integers 1 1.

1. Let s be either 1 or 2, and let k and kj (0 5 j s) be natural numbers less than 6. Suppose that w(P) is a function satisfying w(P) = Cuko(p) O(Xko(logN)-2) with o constant C, and that the function h ( a ) has the property + I(n) = + O(log L))I(n), < for 1 5 j s. 15 of Hua [17]. 6). 8). To show the lower bound for I(n), we appeal to Fourier's inversion formula, and observe that < < where the region of integration is given by the inequalities Xko 5 to 5Xk0,Qj 5 t j 5 5Qj (1 j 5 S ) and n - ( 5 ~ 5 ~x>ot:' ) ~ 5 n - x;.

16) by induction on r , based on Mertens' formula and the recursive formula log t ~ ( rt , ;z ) = c ( ~ / P 7-I ;- 1, P I ) . 14)) completing the proof of the lemma. 3. 1 in mind. Namely, s is either 1 or 2, and the natural numbers k and k j ( 0 5 j 5 s) are less than 6 . 3); C S i ( 9 ,a ) ns;,( 9 ,a ) e ( - a n / q ) , A(q,n) = ~ ( q ) - ~ - ' 8 ( p ,k ) 8 ( p ,k ) + 2, + 1, p e ( ~ 7 k ) > 2 7 and h 2 2, or when p 5 5 and Proof. 1, since k j < 6 (0 5 j 5 s). 3. Assume that ( p ,n). One also has T h e n one has B d ( p ,n ) = B(p,d) Prwf.