New Story

Too Long; Didn't Read

The evaluation of Ξ15 involves detailed proofs and calculations, using lemmas and Cauchy's integral formula to derive essential results.

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

12. Evaluation of Ξ15

In a way similar to the proof of Lemma 8.4, by lemma 8.2 and 5.8, we find that the right side above is equal to

It follows by Cauchy’ integral formula that

Gathering these results together we obtain (12.10). The proof of (12.11) is similar to that of.

Proof. The left side is equal to

Assume |w| = α. In a way similar to the proof of Lemma 12.1, we deduce that

By direct calculation,

and the derivative of

at w = 0 is equal to

This can be written as the form

Since

it follows by simple calculation that

We have