Analytic Number Theory- Jia & Matsumoto by Chaohua Jia, Kohji Matsumoto

Number Theory

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2 of the unpublished lecture note [20] of Iwaniec. 27). 27) to the reader. 72 ANALYTIC NUMBER THEORY Ternary problems i n additive prime number theory Before launching the proofs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 [17] Next we assort basic properties of A d ( q ,n ) et al.

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