**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

## 16. Evaluation of Φ2

Recall that Φ2 is given by (13.9). Write

Similar to (15.3),

where

The following lemma will be proved in Appendix A.

**Lemma 16.2**. *The function*

*is analytic and bounded for σ > 9/10. Further we have*

The contour of integration is moved in the same way as in the proof of Lemma 8.4. Thus the right side above is, by Lemma 16.2, equal to

Hence, by (16.15),

Inserting this into (16.13) and applying Lemma 16.1 we obtain

On the other hand, by Lemma 5.8 and direct calculation,

so that

This together with (16.16) and (16.12) yields

This paper is available on arxiv under CC 4.0 license.