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Discrete Mean Estimates and the Landau-Siegel Zero: Appendix B. Some Arithmetic Sumsby@eigenvalue
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Discrete Mean Estimates and the Landau-Siegel Zero: Appendix B. Some Arithmetic Sums

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June 5th, 2024
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Appendix B delves into proving Lemmas 15.1 and 17.1 concerning arithmetic sums, offering detailed mathematical analysis and proofs for clarity.
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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

Appendix B. Some arithmetic sums

Proof of Lemma 15.1. Put


image


First we claim that


image


Since χ = µ ∗ ν, it follows that


image


Hence


image


This together with Lemma 3.2 yields (B.1).


Next we claim that


image


image


This yields (B.2).


By (B.1) and (B.2), for µ = 2, 3,


image


We proceed to prove theassertion with µ = 2. Since


image


for σ > 1 and


image


it follows that


image


For µ = 1 the proof is therefore reduced to showing that


image


By (4.2) and (4.3), the left side of (B.3) is equal to


image


By a change of variable, for 0.5 ≤ z ≤ 0.504,


image


Hence, in a way similar to the proof of, we find that the left side of (B.3) i


image


Proof of Lemma 17.1. By Lemma 3.1,


image


The sum on the right side is equal to


image


Assume σ > 1. We have


image


If χ(p) = 1, then (see [19, (1.2.10)])


image


if χ(p) = −1, then


image


if χ(p) = 0, then


image


Hence


image


In a way similar to the proof of, by (A) and simple estimate, we find that the integral (14) is equal to the residue of the function


image


at s = 0, plus an acceptable error O, which is equal to


image


This paper is available on arxiv under CC 4.0 license.


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