**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

## Appendix A. Some Euler Products

This appendix is devoted to proving Lemma 8.3, 15.2, 15.3, 16.1 and 16.2. For notational simplicity we shall write

*Proof of Lemma 8.3. Note that*

which are henceforth assumed. We discuss in three cases.

*Case 1. (q, dh) = 1.*

We have

It follows that

This together with the relations

yields (A.1).

*Case 2. q|h.*

We have

so that

This yields (A.3).

This completes the proof.

*Proof of Lemma 16.1. For any q, r, d and l we have*

Hence

and

On the other hand we have

It follows that

It is direct to verify that in either case the assertion holds.

*Proof of Lemma 16.2.* We give a sketch only. If *dl < D, (dl, D)* = 1 and |s − 1| ≤ 5α, then

with

The assertion follows by discussing the cases χ(2) 6= 1 and χ(2) = 1 respectively

This paper is available on arxiv under CC 4.0 license.