By Chaohua Jia, Kohji Matsumoto

Comprises numerous survey articles on best numbers, divisor difficulties, and Diophantine equations, in addition to examine papers on numerous facets of analytic quantity conception difficulties.

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At this stage we benefit from the bilinear error term in Iwaniec's linear sieve. Without it, we can prove Theorems 1 and 2 (i) only with P16and P7,respectively, at present. Thus we require Iwaniec's bilinear error terms within a weighted sieve. This has been made available by Halberstam and Richert [13]. Rather than stating here their result in its general form, we just mention its effect on our particular problem within the proof below. 4) in [13], following Theorem B), by taking U = T = 213, V = 114 and E = 119, for example.

Heath-Brown, Three primes and an almost prime in arithmetic progression. J. London Math. Soc. (2) 23 (l98l), 396-414. [I61 C. Hooley, On a new approach to various problems of Waring 's type. Recent progress in analytic number theory (Durham, 1979), Vol. 1. Academic Press, London-New York, 1981, 127-191. [17] L. K. Hua, Additive Theory of Prime Numbers. Amer. Math. , Providence, Rhode Island, 1965. [18] H. Iwaniec, Primes of the type 4(x, y) form. Acta Arith. 21 (1972), 203-224. A + A, where 4 is a quadratic [19] H.

We proceed to the main objective of this section, and particularly recall the notation f k ( a ;d), gk( a ) , L, M and m defined in $2. In addition to these, we introduce some extra notation which is used throughout this 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.