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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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Eigen Value Equation Population

@eigenvalue

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June 2nd, 2024
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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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Eigen Value Equation Population

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STORY’S CREDIBILITY

Academic Research Paper

Academic Research Paper

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

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

image


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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


image


It follows by Cauchy’ integral formula that


image


image


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


image


Proof. The left side is equal to


image


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


image


By direct calculation,


image


and the derivative of


image


at w = 0 is equal to


image


This can be written as the form


image


Since


image


it follows by simple calculation that


image


We have


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This paper is available on arxiv under CC 4.0 license.


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