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

4 Jun 2024


(1) Yitang Zhang.

  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


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



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


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

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