Mathematical Analysis and Proofs for Evaluating Φ3

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

17. Evaluation of Φ3

Recall that Φ3 is given by (13.10). Write

For σ > 1 we can write

The following lemma will be proved in Appendix B.

Lemma 17.1. We have

Finally, from (17.1), (17.6) and (17.9) we conclude