Discrete Mean Estimates and the Landau-Siegel Zero: Proof of Proposition 2.4

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

10. Proof of Proposition 2.4

The goal of this section is to prove Proposition 2.4. We continue to assume 1 ≤ j ≤ 3.

By Lemma 8.1 we have

The contour of integration is moved in the same way as the proof of Lemma 8.1. Thus, by Lemma 5.8 and 8.2,

This yields (10.8) since the function

Thus, with simple modification, Lemma 10.1 and 10.2 apply to the sums

Noting that

and gathering the above results together we conclude

with

Hence, by Proposition 7.1,