Critical Inequalities and Their Role in Forming the Set Ψ1 in L-Function Theory
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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
3. The set Ψ1
Let ν(n) and υ(n) be given by
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respectively. It is easy to see that
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Lemma 3.1. Assume (A) holds. Then
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Proof. Let
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which has the Euler product representation
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For σ ≥ σ0 > 0, by checking the cases χ(p) = ±1 and χ(p) = 0 respectively, it can be seen that
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and
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the implied constant depending on σ0. Thus φ(s) is analytic for σ > 1/2 and it satisfies
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for σ ≥ σ1 > 1/2, the implied constant depending on σ1. The left side of (3.2) is
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Lemma 3.2. Assume (A) holds. Then we have
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Proof. As the situation is analogous to Lemma 3.1 we give a sketch only. It can be verified that the function
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is analytic for σ > 1/2 and it satisfies
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for σ ≥ σ1 > 1/2, the implies constant depending on σ1. Also, one can verify that
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This completes the proof.
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Lemma 3.3. For any s and any complex numbers c(n) we have
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and
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Proof. The first assertion follows by the orthogonality relation; the second assertion follows by the large sieve inequality.
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Let
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By (3.1) we may write
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By Cauchy’s inequality and the first assertion of Lemma 3.3 we obtain
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Thus we conclude
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Lemma 3.4. The inequality
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Write
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Assume that (A) holds. By Cauchy’s inequality, the second assertion of Lemma 3.2 and Lemma 3.1,
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Thus we conclude
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Lemma 3.5 Assume that (A) holds. The inequality
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Let
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Assume that (A) holds. By Cauchy’s inequality, the first assertion of Lemma 3.2 and Lemma 3.1,
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Thus we conclude
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Lemma 3.6. Assume that (A) holds. The inequality
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We are now in a position to give the definition of Ψ1: Let Ψ1 be the subset of Ψ such that ψ ∈ Ψ1 if and only if the inequalities (3.4), (3.5) and (3.6) simultaneously hold.
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Proposition 2.1 follows from Lemma 3.4, 3.5 and 3.6 immediately.
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This paper is available on arxiv under CC 4.0 license.
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