**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

## 17. Evaluation of Φ3

Recall that Φ3 is given by (13.10). Write

For σ > 1 we can write

The following lemma will be proved in Appendix B.

**Lemma 17.1.** *We have*

Finally, from (17.1), (17.6) and (17.9) we conclude

This paper is available on arxiv under CC 4.0 license.