Discrete Mean Estimates and the Landau-Siegel Zero: Appendix A. Some Euler Products

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

Appendix A. Some Euler Products

This appendix is devoted to proving Lemma 8.3, 15.2, 15.3, 16.1 and 16.2. For notational simplicity we shall write

Proof of Lemma 8.3. Note that

which are henceforth assumed. We discuss in three cases.

Case 1. (q, dh) = 1.

We have

It follows that

This together with the relations

yields (A.1).

Case 2. q|h.

We have

so that

This yields (A.3).

This completes the proof.

Proof of Lemma 16.1. For any q, r, d and l we have

Hence

and

On the other hand we have

It follows that

It is direct to verify that in either case the assertion holds.

Proof of Lemma 16.2. We give a sketch only. If dl < D, (dl, D) = 1 and |s − 1| ≤ 5α, then

with

The assertion follows by discussing the cases χ(2) 6= 1 and χ(2) = 1 respectively