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 15. Evaluation of Φ1 Recall that Φ1 is given by (13.8). In view of (12.2), B(s, ψ) can be written as with Write where Hence First we prove that Since it follows that This yields (15.4). Let κ1(m) be given by Regarding b as an arithmetic function, for σ > 1 we have On the other hand, we can write with It follows by (15.3)-(15.5) and Proposition 14.1 that where The innermost sum above is, by the Mellin transform, equal to where This yields Hence where On substituting n = mk we can writ with Hence it follows that If (q, dl) = 1, then so that for σ > 9/10. In case (q, dl) > 1 and σ > 9/10, the left side above is trivially It follows that the function is analytic and it satisfies for σ > 9/10. The right side of (15.14) can be rewritten as The following lemma will be proved in Appendix B. By (15.19)-(15.21) and Lemma 15.1 we obtain This yields, by (15.21), To apply (15.22) we need two lemmas which will be proved in Appendix A. Lemma 15.2. If |s − 1| < 5α, then Lemma 15.3. For σ ≥ 9/10 the function is analytic and bounded. Further we have By (4.2) and (4.3), By Lemma 15.3, we can move the contour of integration in the same way as in the proof of Lemma 8.4 to obtain This together with Lemma 15.2 and 15.3 yields since It follows by (15.22) that By Lemma 5.8, Hence, by direct calculation, Combining these relations with (15.23) , (15.17) and (15.6) we conclude 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 15. Evaluation of Φ1 Recall that Φ1 is given by (13.8). In view of (12.2), B(s, ψ) can be written as with Write where Hence First we prove that Since it follows that This yields (15.4). Let κ1(m) be given by Regarding b as an arithmetic function, for σ > 1 we have b On the other hand, we can write with It follows by (15.3)-(15.5) and Proposition 14.1 that where The innermost sum above is, by the Mellin transform, equal to where This yields Hence where On substituting n = mk we can writ n = mk with Hence it follows that If ( q, dl ) = 1, then q, dl so that for σ > 9/10. In case (q, dl) > 1 and σ > 9/10, the left side above is trivially It follows that the function is analytic and it satisfies for σ > 9/10. The right side of (15.14) can be rewritten as The following lemma will be proved in Appendix B. By (15.19)-(15.21) and Lemma 15.1 we obtain This yields, by (15.21), To apply (15.22) we need two lemmas which will be proved in Appendix A. Lemma 15.2. If |s − 1| < 5α, then Lemma 15.2. Lemma 15.3. For σ ≥ 9/10 the function Lemma 15.3. For σ ≥ 9/10 the function is analytic and bounded. Further we have is analytic and bounded. Further we have By (4.2) and (4.3), By Lemma 15.3, we can move the contour of integration in the same way as in the proof of Lemma 8.4 to obtain This together with Lemma 15.2 and 15.3 yields since It follows by (15.22) that By Lemma 5.8, Hence, by direct calculation, Combining these relations with (15.23) , (15.17) and (15.6) we conclude 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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Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms

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