Discrete Mean Estimates and the Landau-Siegel Zero: Appendix A. Some Euler Productsby@eigenvalue

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

June 4th, 2024

Appendix A delves into proving Lemmas 8.3, 15.2, 15.3, 16.1, and 16.2 concerning Euler products, offering detailed mathematical analysis and sketches for clarity.

Author:

(1) Yitang Zhang.

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

This paper is available on arxiv under CC 4.0 license.

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