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

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

June 4th, 2024

The proof of Proposition 2.5 involves detailed mathematical analysis, using advanced calculations and propositions like Lemma 8.1 and Proposition 7.1 to establish equations (2.32) and (2.33).

Author:

(1) Yitang Zhang.

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

This paper is available on arxiv under CC 4.0 license.

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