Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ15
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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
12. Evaluation of Ξ15
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In a way similar to the proof of Lemma 8.4, by lemma 8.2 and 5.8, we find that the right side above is equal to
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It follows by Cauchy’ integral formula that
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Gathering these results together we obtain (12.10). The proof of (12.11) is similar to that of.
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Proof. The left side is equal to
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Assume |w| = α. In a way similar to the proof of Lemma 12.1, we deduce that
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By direct calculation,
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and the derivative of
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at w = 0 is equal to
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This can be written as the form
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Since
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it follows by simple calculation that
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We have
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This paper is available on arxiv under CC 4.0 license.
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