Discrete Mean Estimates and the Landau-Siegel Zero: Appendix B. Some Arithmetic Sums
:::info
Author:
(1) Yitang Zhang.
:::
Table of Links
Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
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
\
\
:::info
This paper is available on arxiv under CC 4.0 license.
:::
\
Welcome to Billionaire Club Co LLC, your gateway to a brand-new social media experience! Sign up today and dive into over 10,000 fresh daily articles and videos curated just for your enjoyment. Enjoy the ad free experience, unlimited content interactions, and get that coveted blue check verification—all for just $1 a month!
Account Frozen
Your account is frozen. You can still view content but cannot interact with it.
Please go to your settings to update your account status.
Open Profile Settings