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by Eigen Value Equation Population June 2nd, 2024

**Author:**

(1) Yitang Zhang.

- 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

The goal of this section is to prove Proposition 2.4. We continue to assume 1 ≤ j ≤ 3.

By Lemma 8.1 we have

The contour of integration is moved in the same way as the proof of Lemma 8.1. Thus, by Lemma 5.8 and 8.2,

This yields (10.8) since the function

Thus, with simple modification, Lemma 10.1 and 10.2 apply to the sums

Noting that

and gathering the above results together we conclude

with

Hence, by Proposition 7.1,

This paper is available on arxiv under CC 4.0 license.

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