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Understanding the Role of Analytic Lemmas in Dirichlet L-Functionsby@eigenvalue

Understanding the Role of Analytic Lemmas in Dirichlet L-Functions

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June 2nd, 2024
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The section presents key analytic lemmas, exploring their proofs using techniques like the Mellin transform, partial integration, and the Deuring-Heilbronn phenomenon, to further understand Dirichlet L-functions.
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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

5. Some analytic lemmas

image


image


The proofs of (5.3) and (5.4) are similar.


Lemma 5.2. Let ψ and s be as in Lemma 5.1. Then


image


Proof. The left side is


image


By (2.6) and the Stirling formula, for |w| < 5α,


image


Hence, for 1 ≤ j ≤ 3,


image


The result now follows since


image


Recall that ϑ(s) and ω(s) are given by (2.3) and (2.15) respectively. It is known that


image


For t > 1 we have


image


where


image


Let


image


and


image


Note that


image


image


Proof. By the Mellin transform (see [1], Lemma 2) we have


image


image


with


image


By the relation


image


and Cauchy’s theorem, the proof of (5.8) is reduced to showing that


image


for 1 ≤ j ≤ 5, where Lj denote the segments


image


image


This yields (5.12) with j = 3 .


As a consequence of Lemma 5.3, the Mellin transform


image


is analytic for σ > 0.


Lemma 5.4. (i). If 1/2 ≤ σ ≤ 2, then


image


(ii). If |s − 1| < 10α, then


image


Proof. (i). Using partial integration twice we obtain


image


By (5.10) we have


image


Thus some upper bounds for ∆′′(x) analogous to Lemma 5.3 can be obtained, and (i) follows.


image


Throughout the rest of this paper we assume that (A) holds. This assumption will not be repeated in the statements of the lemmas and propositions in the sequel.


The next two lemmas are weaker forms of the Deuring-Heillbronn Phenomenon.


image


image


Lemma 5.7. We have


image


Proof. The right side of the equality


image


Lemma 5.8. If


image


then


image


where


image


Proof. This follows from the relation


image



(A) and a simple bound for L ′′(w, χ).


image


Proof. It is known that



image


image


so that


image


By (9.1) and the condition |s − ρ| ≫ α for any ρ,


image


so that


image


Further, by (5.16) and Proposition 2.2 (iii),


image


Combining theses estimates we obtain the result. In the case σ < 1/2 the proof is analogous.



This paper is available on arxiv under CC 4.0 license.


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