Author:
(1) Yitang Zhang. 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. We begin by proving some consequences of the inequalities (3.4)-(3.6). Lemma 4.1. Let Proof. By the Stieltjes integral we may write Hence, by partial integration, For G(s, ψ) an entirely analogous bound is valid. The result now follows by (3.4). Lemma 4.2. If s ∈ Ω1, then Proof. We have Thus, similar to (4.2), by partial integration we obtain Lemma 4.3. Let 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 with We may write Since it follows, by changing the order of integration, that Thus the function g(x) is increasing and it satisfies 0 < g(x) < 1. Further we have Note that χψ is a primitive character (modDp). Write so that By (2.4) with θ = ψ and θ = χψ we have This yields, by Stirling’s formula, and Lemma 4.4. Let If s ∈ Ω3, then Proof. By the residue theorem, By (4.2) and (4.3), By (3.5) and partial summation, the second sum on the right side above is On the other hand, by the functional equation, for u = −σ − 1/2, and To prove (4.7) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w), (4.5) and the residue theorem we obtain (4.7). To prove (4.8) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w) and (4.5) we see that the left side of (4.8) is with s ∗ = 1 + α − s¯. By partial integration, The estimate (4.9) follows by moving the contour of integration to the vertical segments and to the two connecting horizontal segments and applying (4.5) and trivial bounds for ω1(w) and the involved sum. In order to prove Proposition 2.2, it is appropriate to deal with the function 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 by Lemma 4.1 and 4.2. This together with Lemma 4.4 implies that for s ∈ Ω3, where The proof of Proposition 2.2 is reduced to proving three lemmas as follows. Lemma 4.5. If then Proof. We discuss in two cases. By Lemma 4.2 and trivial estimation, Hence, by (4.5), The result now follows by (4.10). Since |B(1/2 + it, ψ)| = 1, it follows that Hence, by (4.10), Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying 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 In either case (4.16) holds. 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 Proof. In a way similar to the proof of Lemma 4.6, it is direct to verify that We conclude this section by giving a result which is implied in the proof of Proposition 2.2. Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have Proof. It follows from Lemma 4.4 that This paper is available on arxiv under CC 4.0 license. Author: (1) Yitang Zhang. Author: Author: (1) Yitang Zhang. 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 Abstract & Introduction Abstract & Introduction Notation and outline of the proof Notation and outline of the proof The set Ψ1 The set Ψ1 Zeros of L(s, ψ)L(s, χψ) in Ω Zeros of L(s, ψ)L(s, χψ) in Ω Some analytic lemmas Some analytic lemmas Approximate formula for L(s, ψ) Approximate formula for L(s, ψ) Mean value formula I Mean value formula I Evaluation of Ξ11 Evaluation of Ξ11 Evaluation of Ξ12 Evaluation of Ξ12 Proof of Proposition 2.4 Proof of Proposition 2.4 Proof of Proposition 2.6 Proof of Proposition 2.6 Evaluation of Ξ15 Evaluation of Ξ15 Approximation to Ξ14 Approximation to Ξ14 Mean value formula II Mean value formula II Evaluation of Φ1 Evaluation of Φ1 Evaluation of Φ2 Evaluation of Φ2 Evaluation of Φ3 Evaluation of Φ3 Proof of Proposition 2.5 Proof of Proposition 2.5 Appendix A. Some Euler products Appendix A. Some Euler products Appendix B. Some arithmetic sums Appendix B. Some arithmetic sums References 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. We begin by proving some consequences of the inequalities (3.4)-(3.6). Lemma 4.1. Let Lemma 4.1. Let Proof . By the Stieltjes integral we may write Proof Hence, by partial integration, For G(s, ψ) an entirely analogous bound is valid. The result now follows by (3.4). Lemma 4.2. If s ∈ Ω1, then Lemma 4.2. If s ∈ Ω1, then Proof . We have Proof Thus, similar to (4.2), by partial integration we obtain Lemma 4.3. Let Lemma 4.3. Let 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 with We may write Since it follows, by changing the order of integration, that Thus the function g(x) is increasing and it satisfies 0 < g(x) < 1. Further we have Note that χψ is a primitive character (modDp). Write so that By (2.4) with θ = ψ and θ = χψ we have This yields, by Stirling’s formula, and Lemma 4.4. Let Lemma 4.4. Let If s ∈ Ω3, then If s ∈ Ω3, then Proof . By the residue theorem, Proof By (4.2) and (4.3), By (3.5) and partial summation, the second sum on the right side above is On the other hand, by the functional equation, for u = −σ − 1/2, and To prove (4.7) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w), (4.5) and the residue theorem we obtain (4.7). To prove (4.8) we move the contour of integration to the vertical segments and to the two connecting horizontal segments By a trivial bound for ω1(w) and (4.5) we see that the left side of (4.8) is with s ∗ = 1 + α − s¯. By partial integration, The estimate (4.9) follows by moving the contour of integration to the vertical segments and to the two connecting horizontal segments and applying (4.5) and trivial bounds for ω1(w) and the involved sum. In order to prove Proposition 2.2, it is appropriate to deal with the function 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 by Lemma 4.1 and 4.2. This together with Lemma 4.4 implies that for s ∈ Ω3, where The proof of Proposition 2.2 is reduced to proving three lemmas as follows. Lemma 4.5. If Lemma 4.5. If then Proof . We discuss in two cases. Proof By Lemma 4.2 and trivial estimation, Hence, by (4.5), The result now follows by (4.10). Since |B(1/2 + it, ψ)| = 1, it follows that Hence, by (4.10), Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying 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 In either case (4.16) holds. 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 Proof . In a way similar to the proof of Lemma 4.6, it is direct to verify that Proof We conclude this section by giving a result which is implied in the proof of Proposition 2.2. Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have Proof . It follows from Lemma 4.4 that Proof This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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