Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ11

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

8. Evaluation of Ξ11

We first prove a general result as follows.

By Proposition 7.1, our goal is reduced to evaluating the sum

Write

so that

Lemma 8.2. Suppose T < x < P. Then for µ = 6, 7

where

Proof. The sum is equal to

We move the contour of integration to the vertical segments

and to the two connecting horizontal segments

It follows by Lemma 5.6 that

The result now follows by direct calculation.

Combining these results with Lemma 8.3, we find that the integral (8.9) is equal to

The result now follows by direct calculation.