Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ11by@eigenvalue

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

June 2nd, 2024

This section dives into the calculation of Ξ11, employing Lemma 8.2 and Lemma 8.3 to evaluate integrals and apply the large sieve inequality for a comprehensive understanding.

Author:

(1) Yitang Zhang.

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.

This paper is available on arxiv under CC 4.0 license.

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