**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

## 13. Approximation to Ξ14

In this section we establish an approximation to Ξ14.

Assume that ψ ∈ Ψ1 and ρ ∈ Z(ψ). By Lemma 5.2 and (2.2),

By Lemma 6.1,

and, by Lemma 5.1,

Hence

By Lemma 6.1 and 5.1,

We insert this into (13.2) and then insert the result into (13.1). Thus we obtain

where

where

Inserting this into (12.4) we obtain

Combining (2.34), Cauchy’s inequality, Proposition 7.1, Lemma 5.9, 6.1 and 3.3, we can verify that

For example, by (2.34)

the right side being estimated via Lemma 5.9, 6.1 and 3.3

This paper is available on arxiv under CC 4.0 license.