**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

## 5. Some analytic lemmas

The proofs of (5.3) and (5.4) are similar.

**Lemma 5.2.** *Let ψ and s be as in Lemma 5.1. Then*

*Proof*. The left side is

By (2.6) and the Stirling formula, for |w| < 5α,

Hence, for 1 ≤ j ≤ 3,

The result now follows since

Recall that ϑ(s) and ω(s) are given by (2.3) and (2.15) respectively. It is known that

For t > 1 we have

where

Let

and

Note that

*Proof*. By the Mellin transform (see [1], Lemma 2) we have

with

By the relation

and Cauchy’s theorem, the proof of (5.8) is reduced to showing that

for 1 ≤ j ≤ 5, where Lj denote the segments

This yields (5.12) with j = 3 .

As a consequence of Lemma 5.3, the Mellin transform

is analytic for σ > 0.

**Lemma 5.4**. (i). *If 1/2 ≤ σ ≤ 2, then*

(ii). If |s − 1| < 10α, *then*

*Proof*. (i). Using partial integration twice we obtain

By (5.10) we have

Thus some upper bounds for ∆′′(x) analogous to Lemma 5.3 can be obtained, and (i) follows.

*Throughout the rest of this paper we assume that (A) holds. This assumption will not be repeated in the statements of the lemmas and propositions in the sequel.*

The next two lemmas are weaker forms of the Deuring-Heillbronn Phenomenon.

**Lemma 5.7.** *We have*

*Proof*. The right side of the equality

**Lemma 5.8.** *If*

*then*

*where*

*Proof*. This follows from the relation

(A) and a simple bound for L ′′(w, χ).

*Proof*. It is known that

so that

By (9.1) and the condition |s − ρ| ≫ α for any ρ,

so that

Further, by (5.16) and Proposition 2.2 (iii),

Combining theses estimates we obtain the result. In the case σ < 1/2 the proof is analogous.

This paper is available on arxiv under CC 4.0 license.