**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

## 14. Mean-value formula II

Recall that we always assume ψ is a primitive character (mod p), p ∼ P. Sometimes we write pψ for the modulus p.

Let k ∗ = {κ ∗ (m)} and a ∗ = {a ∗ (n)} denote sequences of complex numbers satisfying

The goal of this section is to prove

Proposition 14.1. Suppose |β| < 5α. Then

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