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by Eigen Value Equation Population June 5th, 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

*Proof of Lemma 15.1.* Put

First we claim that

Since χ = µ ∗ ν, it follows that

Hence

This together with Lemma 3.2 yields (B.1).

Next we claim that

This yields (B.2).

By (B.1) and (B.2), for µ = 2, 3,

We proceed to prove theassertion with µ = 2. Since

for σ > 1 and

it follows that

For µ = 1 the proof is therefore reduced to showing that

By (4.2) and (4.3), the left side of (B.3) is equal to

By a change of variable, for 0.5 ≤ z ≤ 0.504,

Hence, in a way similar to the proof of, we find that the left side of (B.3) i

*Proof of Lemma 17.1.* By Lemma 3.1,

The sum on the right side is equal to

Assume σ > 1. We have

If χ(p) = 1, then (see [19, (1.2.10)])

if χ(p) = −1, then

if χ(p) = 0, then

Hence

In a way similar to the proof of, by (A) and simple estimate, we find that the integral (14) is equal to the residue of the function

at *s* = 0, plus an acceptable error *O*, which is equal to

This paper is available on arxiv under CC 4.0 license.

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