Discrete Mean Estimates and the Landau-Siegel Zero: Appendix B. Some Arithmetic Sums

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

Appendix B. Some arithmetic sums

Proof of Lemma 15.1. Put

First we claim that

Since χ = µ ∗ ν, it follows that

Hence

This together with Lemma 3.2 yields (B.1).

Next we claim that

This yields (B.2).

By (B.1) and (B.2), for µ = 2, 3,

We proceed to prove theassertion with µ = 2. Since

for σ > 1 and

it follows that

For µ = 1 the proof is therefore reduced to showing that

By (4.2) and (4.3), the left side of (B.3) is equal to

By a change of variable, for 0.5 ≤ z ≤ 0.504,

Hence, in a way similar to the proof of, we find that the left side of (B.3) i

Proof of Lemma 17.1. By Lemma 3.1,

The sum on the right side is equal to

Assume σ > 1. We have

If χ(p) = 1, then (see [19, (1.2.10)])

if χ(p) = −1, then

if χ(p) = 0, then

Hence

In a way similar to the proof of, by (A) and simple estimate, we find that the integral (14) is equal to the residue of the function

at s = 0, plus an acceptable error O, which is equal to