Discrete Mean Estimates and the Landau-Siegel Zero: Proof of Proposition 2.5

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

18. Proof of Proposition 2.5

By the discussion at the end of Section 2, it suffices to prove (2.32) and (2.33).

Proof of (2.32).

By (12.3), (12,17), (13.7), (15.24), (16.17) and (17.10),

In view of (15.), we can write

By calculation (there is a theoretical interpretation),

Hence

Direct calculation shows that

It follows from (8.24), (9.8) and (18.2) that

This with together (8.23), (9.7) and (18.1) yields (2.32).

Proof of (2.33).

By Lemma 8.1,

We have

The right side is split into three sums according to

Thus we have the crude bound

so that

This yields (2.33) by Lemma 8.1 and Proposition 7.1