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121(1985), 319-350. [6] D. A. Goldston, J. Pintz and C. Y. Yildirim, Primes in Tuples I, Ann. Math. 170(2009), 819-862. [7] D. A. Goldston, J. Pintz and C. Y. Yildirim, Primes in Tuples II, Acta Math. 204(2010), 1-47. [8] D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity, Canad. J. Math. 34(1982), 1365-1377. [9] D. R. Heath-Brown, The divisor function d3 (n) in arithmetic progressions, Acta Arith. 47(1986), 29-56. [10] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith.
6), this leads to Qi (d1 , k) d1+ε 1 krN3 . 47 N31+ε by Lemma 9. 2). 14) is x1− +ε M −1 if k < 2K. 7) is reduced to showing that 1 K N (d1 , k) x1− /2+ε M −1 . 16) k∼K 14. 16) that will complete the proof of Theorem 2. We start with the relation h(n2 + hkr) ≡ l + kr (mod d1 ) ¯ 2 (mod d1 ). Thus we may rewrite N (d1 , k) for (h, d1 ) = (n2 + hkr, d1 ) = 1, where l ≡ hn as N (d1 , k) = ν(l; d1 ) ed1 b(l + kr)n3 l( mod d1 ) (l+kr,d1 )=1 n3 N3 (n3 ,d1 )=1 with fˆ(h/d1 ). ν(l; d1 ) = ¯ 2 ≡l(d1 ) hn Here is restriction to 1 ≤ h < H, (h, d1 ) = 1 and n2 inequality that N2 .
Namely we have S2 (r, a) = q1 b1 (q2 ,q1 )=1 × b2 c(r, a; q1 , b1 )c(r, a; q2 , b2 ) q2 ϕ(r) f (m) + O(xN D0−1 R−2 LB ). 3) By Lemma 7, for (n1 , q1 r) = 1 we have f (m) = mn1 ≡a(r) mn1 ≡b1 (q1 ) 1 q1 r h eq1 r (−hµ) + O(x−2 ), q1 r fˆ |h|