Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ15

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


12. Evaluation of Ξ15

In a way similar to the proof of Lemma 8.4, by lemma 8.2 and 5.8, we find that the right side above is equal to

It follows by Cauchy’ integral formula that

Gathering these results together we obtain (12.10). The proof of (12.11) is similar to that of.

Proof. The left side is equal to

Assume |w| = α. In a way similar to the proof of Lemma 12.1, we deduce that

By direct calculation,

and the derivative of

at w = 0 is equal to

This can be written as the form


it follows by simple calculation that

We have

This paper is available on arxiv under CC 4.0 license.