Analytic Number Theory- Jia & Matsumoto by Chaohua Jia, Kohji Matsumoto
By Chaohua Jia, Kohji Matsumoto
Contains numerous survey articles on major numbers, divisor difficulties, and Diophantine equations, in addition to examine papers on quite a few points of analytic quantity conception difficulties.
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2 of the unpublished lecture note  of Iwaniec. 27). 27) to the reader. 72 ANALYTIC NUMBER THEORY Ternary problems i n additive prime number theory Before launching the proofs of Theorems 1 and 2 (i), we record a simple fact as a lemma. 1. Let v(n) be the number of distinct prime divisors of n, and A be any constant exceeding 3. Then one has v(n) 2Aloglog N for all but O(N(log N ) - ~ )natural numbers n 5 N . < 73 [N, (6/5)N] where N1 is the set defined in the statement of Theorem 2. Also we say simply "for almost all n" instead of "for all n E N(kl) with at most O(N(1og N ) - ~ )possible exceptions", within the current section.
We define the intervals denote by rt(Q) the union of all rt(q,a; Q) with 0 5 a I q I Q and (q, a) = 1, and write n(Q) = [O, 11 \ rt(Q), for a positive number Q. Note that the intervals rt(q, a ; Q) composing rt(Q) are pairwise disjoint provided that Q 5 X2. For the interest of saving space, we also introduce the notation When d $ D but d $ V, we have where (a,) is an arbitrary sequence satisfying 58 Ternary problems in additive prime number theory A N A L Y T I C NUMBER T H E O R Y - for all x Xk.
1. Let B(p, k ) be the number such that power of p dividing k , and let (iii) A d ( p ,n ) = A ( p , n ) = 0 , when p h 5. 49 d ) k ) , which For h 5 k , we may observe that s k ( p h ,a d k ) = s k ( p h , gives A ~ (n ) ~= A ~ ( ,~ ,( p~h), n). 4). Next we have is the highest when p = 2 and k is even, when p > 2 or k is odd. 3) + T h e n one has S; ( p h ,a ) = 0 when p a and h > y ( p , k ) . But, when 1 5 h 5 k , we see Proof. 3 of Hua  Next we assort basic properties of A d ( q ,n ) et al.