Author:
(1) Yitang Zhang.
Appendix A. Some Euler products
Appendix B. Some arithmetic sums
Let ν(n) and υ(n) be given by
respectively. It is easy to see that
Lemma 3.1. Assume (A) holds. Then
Proof. Let
which has the Euler product representation
For σ ≥ σ0 > 0, by checking the cases χ(p) = ±1 and χ(p) = 0 respectively, it can be seen that
and
the implied constant depending on σ0. Thus φ(s) is analytic for σ > 1/2 and it satisfies
for σ ≥ σ1 > 1/2, the implied constant depending on σ1. The left side of (3.2) is
Lemma 3.2. Assume (A) holds. Then we have
Proof. As the situation is analogous to Lemma 3.1 we give a sketch only. It can be verified that the function
is analytic for σ > 1/2 and it satisfies
for σ ≥ σ1 > 1/2, the implies constant depending on σ1. Also, one can verify that
This completes the proof.
Lemma 3.3. For any s and any complex numbers c(n) we have
and
Proof. The first assertion follows by the orthogonality relation; the second assertion follows by the large sieve inequality.
Let
By (3.1) we may write
By Cauchy’s inequality and the first assertion of Lemma 3.3 we obtain
Thus we conclude
Lemma 3.4. The inequality
Write
Assume that (A) holds. By Cauchy’s inequality, the second assertion of Lemma 3.2 and Lemma 3.1,
Thus we conclude
Lemma 3.5 Assume that (A) holds. The inequality
Let
Assume that (A) holds. By Cauchy’s inequality, the first assertion of Lemma 3.2 and Lemma 3.1,
Thus we conclude
Lemma 3.6. Assume that (A) holds. The inequality
We are now in a position to give the definition of Ψ1: Let Ψ1 be the subset of Ψ such that ψ ∈ Ψ1 if and only if the inequalities (3.4), (3.5) and (3.6) simultaneously hold.
Proposition 2.1 follows from Lemma 3.4, 3.5 and 3.6 immediately.