Detailed Lemmas on Zeros of Dirichlet L-Functions in Ω
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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
4. Zeros of L(s, ψ)L(s, χψ) in Ω
In this section we prove Proposition 2.2. We henceforth assume that ψ(mod p) ∈ Ψ1. This assumption will not be repeated in the statements of Lemma 4.1-4.8.
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We begin by proving some consequences of the inequalities (3.4)-(3.6).
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Lemma 4.1. Let
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Proof. By the Stieltjes integral we may write
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Hence, by partial integration,
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For G(s, ψ) an entirely analogous bound is valid. The result now follows by (3.4).
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Lemma 4.2. If s ∈ Ω1, then
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Proof. We have
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Thus, similar to (4.2), by partial integration we obtain
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Lemma 4.3. Let
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We proceed to establish an approximate formula for L(s, ψ)L(s, χψ). For this purpose we first introduce a weight g(x) that will find application at various places. Let
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with
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We may write
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Since
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it follows, by changing the order of integration, that
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Thus the function g(x) is increasing and it satisfies 0 < g(x) < 1. Further we have
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Note that χψ is a primitive character (modDp). Write
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so that
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By (2.4) with θ = ψ and θ = χψ we have
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This yields, by Stirling’s formula,
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and
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Lemma 4.4. Let
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If s ∈ Ω3, then
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Proof. By the residue theorem,
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By (4.2) and (4.3),
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By (3.5) and partial summation, the second sum on the right side above is
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On the other hand, by the functional equation, for u = −σ − 1/2,
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and
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To prove (4.7) 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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By a trivial bound for ω1(w), (4.5) and the residue theorem we obtain (4.7).
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To prove (4.8) 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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By a trivial bound for ω1(w) and (4.5) we see that the left side of (4.8) is
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with s ∗ = 1 + α − s¯. By partial integration,
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The estimate (4.9) follows by moving the contour of integration to the vertical segments
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and to the two connecting horizontal segments
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and applying (4.5) and trivial bounds for ω1(w) and the involved sum.
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In order to prove Proposition 2.2, it is appropriate to deal with the function
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By Lemma 4.2, A(s, ψ) is analytic and it has the same zeros as L(s, ψ)L(s, ψχ) in Ω1. Further, for s ∈ Ω1, we have
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by Lemma 4.1 and 4.2. This together with Lemma 4.4 implies that
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for s ∈ Ω3, where
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The proof of Proposition 2.2 is reduced to proving three lemmas as follows.
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Lemma 4.5. If
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then
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Proof. We discuss in two cases.
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By Lemma 4.2 and trivial estimation,
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Hence, by (4.5),
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The result now follows by (4.10).
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Since |B(1/2 + it, ψ)| = 1, it follows that
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Hence, by (4.10),
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Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying
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Proof. It suffices to show that the function A(1/2 + iγ + w, ψ) has exactly one zero inside the circle |w| = α(1 − c ′αL), counted with multiplicity. By the Rouch´e theorem, this can be reduced to proving that
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In either case (4.16) holds.
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Lemma 4.5 and 4.6 together imply the assertions (i) and (ii) of Proposition 2.2. It is also proved that the gap between any distinct zeros of A(s, ψ) in Ω is > α(1 − c ′αL). To complete the proof of the gap assertion (iii), it now suffices to prove
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Proof. In a way similar to the proof of Lemma 4.6, it is direct to verify that
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We conclude this section by giving a result which is implied in the proof of Proposition 2.2.
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Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have
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Proof. It follows from Lemma 4.4 that
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This paper is available on arxiv under CC 4.0 license.
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