Discrete Mean Estimates and the Landau-Siegel Zero: Approximation to Ξ14

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

13. Approximation to Ξ14

In this section we establish an approximation to Ξ14.

Assume that ψ ∈ Ψ1 and ρ ∈ Z(ψ). By Lemma 5.2 and (2.2),

By Lemma 6.1,

and, by Lemma 5.1,

Hence

By Lemma 6.1 and 5.1,

We insert this into (13.2) and then insert the result into (13.1). Thus we obtain

where

where

Inserting this into (12.4) we obtain

Combining (2.34), Cauchy’s inequality, Proposition 7.1, Lemma 5.9, 6.1 and 3.3, we can verify that

For example, by (2.34)

the right side being estimated via Lemma 5.9, 6.1 and 3.3