**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

## 6. Approximate formula for L(s, Ïˆ)

Write

Let

**Lemma 6.1.** *Suppose* Ïˆ(mod p) âˆˆ Î¨, |Ïƒ âˆ’ 1/2| < 2Î± and |t âˆ’ 2Ï€t0| < L1 + 2. Then

L(s, Ïˆ) = K(s, Ïˆ) + Z(s, Ïˆ)N(1 âˆ’ s, ÏˆÂ¯) + O(E1(s, Ïˆ)),

where

*and where*

*Proof*. By (4.3) we have

The left side above is, by moving the line of integration to u = âˆ’1, equal to

It therefore suffices to show that

For u = âˆ’1 we have, by the functional equation (2.2) with Î¸ = Ïˆ,

We first show that

We move the contour of integration in (6.2) to the vertical segments

and

with the horizontal connecting segments

whence (6.2) follows. The proof of (6.1) is therefore reduced to showing that

This paper is available on arxiv under CC 4.0 license.