Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ11
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Author:
(1) Yitang Zhang.
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Table of Links
Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
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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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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This paper is available on arxiv under CC 4.0 license.
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