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Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ15by@eigenvalue

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

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The evaluation of Ξ15 involves detailed proofs and calculations, using lemmas and Cauchy's integral formula to derive essential results.
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Author:

(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

References

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



Since



it follows by simple calculation that



We have



This paper is available on arxiv under CC 4.0 license.