Table of Links

1. Abstract & Introduction
2. Notation and outline of the proof
3. The set Ψ1
4. Zeros of L(s, ψ)L(s, χψ) in Ω
5. Some analytic lemmas
6. Approximate formula for L(s, ψ)
7. Mean value formula I
8. Evaluation of Ξ11
9. Evaluation of Ξ12
10. Proof of Proposition 2.4
11. Proof of Proposition 2.6
12. Evaluation of Ξ15
13. Approximation to Ξ14
14. Mean value formula II
15. Evaluation of Φ1
16. Evaluation of Φ2
17. Evaluation of Φ3
18. Proof of Proposition 2.5

Appendix A. Some Euler products

Appendix B. Some arithmetic sums

References

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.