Mathematical Proofs for Evaluating Φ2

3 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


16. Evaluation of Φ2

Recall that Φ2 is given by (13.9). Write

Similar to (15.3),


The following lemma will be proved in Appendix A.

Lemma 16.2. The function

is analytic and bounded for σ > 9/10. Further we have

The contour of integration is moved in the same way as in the proof of Lemma 8.4. Thus the right side above is, by Lemma 16.2, equal to

Hence, by (16.15),

Inserting this into (16.13) and applying Lemma 16.1 we obtain

On the other hand, by Lemma 5.8 and direct calculation,

so that

This together with (16.16) and (16.12) yields

This paper is available on arxiv under CC 4.0 license.