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Discrete Mean Estimates and the Landau-Siegel Zero: Approximation to Ξ14by@eigenvalue
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Discrete Mean Estimates and the Landau-Siegel Zero: Approximation to Ξ14

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Eigen Value Equation Population

@eigenvalue

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June 2nd, 2024
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The approximation to Ξ14 is established through a series of lemmas, propositions, and Cauchy's inequality, integrating results and proofs from various sections.
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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

13. Approximation to Ξ14

In this section we establish an approximation to Ξ14.


Assume that ψ ∈ Ψ1 and ρ ∈ Z(ψ). By Lemma 5.2 and (2.2),


image


By Lemma 6.1,


image


and, by Lemma 5.1,


image


Hence


image


By Lemma 6.1 and 5.1,


image


We insert this into (13.2) and then insert the result into (13.1). Thus we obtain


image


where


image


image


where


image


Inserting this into (12.4) we obtain


image


Combining (2.34), Cauchy’s inequality, Proposition 7.1, Lemma 5.9, 6.1 and 3.3, we can verify that


image


For example, by (2.34)


image


the right side being estimated via Lemma 5.9, 6.1 and 3.3


This paper is available on arxiv under CC 4.0 license.


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