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Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ11by@eigenvalue

Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ11

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Eigen Value Equation Population

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June 2nd, 2024
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This section dives into the calculation of Ξ11, employing Lemma 8.2 and Lemma 8.3 to evaluate integrals and apply the large sieve inequality for a comprehensive understanding.
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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

8. Evaluation of Ξ11

We first prove a general result as follows.


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image


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By Proposition 7.1, our goal is reduced to evaluating the sum


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Write


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so that


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Lemma 8.2. Suppose T < x < P. Then for µ = 6, 7


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where


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Proof. The sum is equal to


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We move the contour of integration to the vertical segments


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and to the two connecting horizontal segments


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It follows by Lemma 5.6 that


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The result now follows by direct calculation.


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Combining these results with Lemma 8.3, we find that the integral (8.9) is equal to


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The result now follows by direct calculation.


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image


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This paper is available on arxiv under CC 4.0 license.


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