Mathematical Proofs for Evaluating Φ2

Table of Links

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

References

16. Evaluation of Φ2

Recall that Φ2 is given by (13.9). Write

Similar to (15.3),

where

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