Sums and differences of quartic norms

Chen's double sieve, Goldbach's conjecture and the twin prime problem

“…The following result is an immediate consequence of Corollary 2 of [11], Lemma 6 of [12] and the proposition of [18]. Lemma 2.5.…”

“…, Lemma 2.5 with the choice 18 , where λ + l (q) is well factorisable of order 1 and of level Q = x θ(t 2 ) with θ(t 2 ) = (2 + t 2 )/4.…”

“…λ + l (q) |B(t 1 , t 2 , t 3 ) q | − |B(t 1 , t 2 , t 3 )| ϕ(q) ≪ x (log x) 18 , where λ + l (q) is well factorable of order 1 and of level Q = x θ(t2) with θ(t 2 ) = (2 + t 2 )/4. Thus we find by (2.4) [For the integrals F 13 and F 14 , we make use of ω(u) 0.561522 for u 3.5 and ω(u) 0.567144 for u 2.]…”

“…(ii) Certainly we could obtain a better constant than 3.3996 in Theorem 3 if we used mean value theorems of ( [11], Corollary 2), ( [12], Lemma 6) and ( [18], Proposition) as in the proof of Theorem 4. But the numerical computation involved would be quite complicated.…”

“…The value (6 − 5ν)/10 in [ε, 1/15] comes from Corollary 2 (ii) of [11]. The intervals [1/15, 1/10] and [1/10, 3/14] follow from the proposition of [18] by decomposing λ = λ 1 * λ 2 with…”