Discrete Mean Estimates and the Landau-Siegel Zero: Proof of Proposition 2.4

2 Jun 2024


(1) Yitang Zhang.

  1. Abstract & Introduction
  2. Notation and outline of the proof
  3. The set Ψ1
  4. Zeros of L(s, ψ)L(s, χψ) in Ω
  5. Some analytic lemmas
  6. Approximate formula for L(s, ψ)
  7. Mean value formula I
  8. Evaluation of Ξ11
  9. Evaluation of Ξ12
  10. Proof of Proposition 2.4
  11. Proof of Proposition 2.6
  12. Evaluation of Ξ15
  13. Approximation to Ξ14
  14. Mean value formula II
  15. Evaluation of Φ1
  16. Evaluation of Φ2
  17. Evaluation of Φ3
  18. Proof of Proposition 2.5

Appendix A. Some Euler products

Appendix B. Some arithmetic sums


10. Proof of Proposition 2.4

The goal of this section is to prove Proposition 2.4. We continue to assume 1 ≤ j ≤ 3.

By Lemma 8.1 we have

The contour of integration is moved in the same way as the proof of Lemma 8.1. Thus, by Lemma 5.8 and 8.2,

This yields (10.8) since the function

Thus, with simple modification, Lemma 10.1 and 10.2 apply to the sums

Noting that

and gathering the above results together we conclude


Hence, by Proposition 7.1,

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