Author:
(1) Yitang Zhang. Table of Links Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5 Appendix A. Some Euler products Appendix B. Some arithmetic sums References Appendix A. Some Euler Products This appendix is devoted to proving Lemma 8.3, 15.2, 15.3, 16.1 and 16.2. For notational simplicity we shall write Proof of Lemma 8.3. Note that which are henceforth assumed. We discuss in three cases. Case 1. (q, dh) = 1. We have It follows that This together with the relations yields (A.1). Case 2. q|h. We have so that This yields (A.3). This completes the proof. Proof of Lemma 16.1. For any q, r, d and l we have Hence and On the other hand we have It follows that It is direct to verify that in either case the assertion holds. Proof of Lemma 16.2. We give a sketch only. If dl < D, (dl, D) = 1 and |s − 1| ≤ 5α, then with The assertion follows by discussing the cases χ(2) 6= 1 and χ(2) = 1 respectively This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. Table of Links Abstract & Introduction Notation and outline of the proof The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Approximate formula for L(s, ψ) Mean value formula I Evaluation of Ξ11 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.6 Evaluation of Ξ15 Approximation to Ξ14 Mean value formula II Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ3 Proof of Proposition 2.5 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References References Appendix A. Some Euler Products This appendix is devoted to proving Lemma 8.3, 15.2, 15.3, 16.1 and 16.2. For notational simplicity we shall write Proof of Lemma 8.3. Note that Proof of Lemma 8.3. Note that which are henceforth assumed. We discuss in three cases. Case 1. (q, dh) = 1. Case 1. (q, dh) = 1. We have It follows that This together with the relations yields (A.1). Case 2. q|h. Case 2. q|h. We have so that This yields (A.3). This completes the proof. Proof of Lemma 16.1. For any q, r, d and l we have Proof of Lemma 16.1. For any q, r, d and l we have Hence and On the other hand we have It follows that It is direct to verify that in either case the assertion holds. Proof of Lemma 16.2. We give a sketch only. If dl < D, (dl, D) = 1 and |s − 1| ≤ 5α, then Proof of Lemma 16.2. dl < D, (dl, D) with The assertion follows by discussing the cases χ(2) 6= 1 and χ(2) = 1 respectively This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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