Discrete Mean Estimates and the Landau-Siegel Zero: Proof of Proposition 2.5

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 18. Proof of Proposition 2.5 By the discussion at the end of Section 2, it suffices to prove (2.32) and (2.33). Proof of (2.32). By (12.3), (12,17), (13.7), (15.24), (16.17) and (17.10), In view of (15.), we can write By calculation (there is a theoretical interpretation), Hence Direct calculation shows that It follows from (8.24), (9.8) and (18.2) that This with together (8.23), (9.7) and (18.1) yields (2.32). Proof of (2.33). By Lemma 8.1, We have The right side is split into three sums according to Thus we have the crude bound so that This yields (2.33) by Lemma 8.1 and Proposition 7.1 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 18. Proof of Proposition 2.5 By the discussion at the end of Section 2, it suffices to prove (2.32) and (2.33). Proof of (2.32). Proof of (2.32). By (12.3), (12,17), (13.7), (15.24), (16.17) and (17.10), In view of (15.), we can write By calculation (there is a theoretical interpretation), Hence Direct calculation shows that It follows from (8.24), (9.8) and (18.2) that This with together (8.23), (9.7) and (18.1) yields (2.32). Proof of (2.33). Proof of (2.33). By Lemma 8.1, We have The right side is split into three sums according to Thus we have the crude bound so that This yields (2.33) by Lemma 8.1 and Proposition 7.1 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