**Author:**

(1) Yitang Zhang.

## Table of Links

- Abstract & Introduction
- Notation and outline of the proof
- The set Î¨1
- Zeros of L(s, Ïˆ)L(s, Ï‡Ïˆ) in â„¦
- Some analytic lemmas
- Approximate formula for L(s, Ïˆ)
- Mean value formula I
- Evaluation of Îž11
- Evaluation of Îž12
- Proof of Proposition 2.4
- Proof of Proposition 2.6
- Evaluation of Îž15
- Approximation to Îž14
- Mean value formula II
- Evaluation of Î¦1
- Evaluation of Î¦2
- Evaluation of Î¦3
- 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 Ïˆ is a primitive character (mod p), p âˆ¼ P. Sometimes we write pÏˆ for the modulus p.

Let k âˆ— = {Îº âˆ— (m)} and a âˆ— = {a âˆ— (n)} denote sequences of complex numbers satisfying

The goal of this section is to prove

Proposition 14.1. Suppose |Î²| < 5Î±. Then

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