**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

## Appendix B. Some arithmetic sums

*Proof of Lemma 15.1.* Put

First we claim that

Since Ï‡ = Âµ âˆ— Î½, it follows that

Hence

This together with Lemma 3.2 yields (B.1).

Next we claim that

This yields (B.2).

By (B.1) and (B.2), for Âµ = 2, 3,

We proceed to prove theassertion with Âµ = 2. Since

for Ïƒ > 1 and

it follows that

For Âµ = 1 the proof is therefore reduced to showing that

By (4.2) and (4.3), the left side of (B.3) is equal to

By a change of variable, for 0.5 â‰¤ z â‰¤ 0.504,

Hence, in a way similar to the proof of, we find that the left side of (B.3) i

*Proof of Lemma 17.1.* By Lemma 3.1,

The sum on the right side is equal to

Assume Ïƒ > 1. We have

If Ï‡(p) = 1, then (see [19, (1.2.10)])

if Ï‡(p) = âˆ’1, then

if Ï‡(p) = 0, then

Hence

In a way similar to the proof of, by (A) and simple estimate, we find that the integral (14) is equal to the residue of the function

at *s* = 0, plus an acceptable error *O*, which is equal to

This paper is available on arxiv under CC 4.0 license.