paint-brush
Detailed Lemmas on Zeros of Dirichlet L-Functions in Ωby@eigenvalue

Detailed Lemmas on Zeros of Dirichlet L-Functions in Ω

by Eigen Value Equation Population
Eigen Value Equation Population  HackerNoon profile picture

Eigen Value Equation Population

@eigenvalue

We cover research, technology, & documentation about special scalar values...

June 2nd, 2024
Read on Terminal Reader
Read this story in a terminal
Print this story
Read this story w/o Javascript
Read this story w/o Javascript
tldt arrow

Too Long; Didn't Read

This section proves Proposition 2.2, establishing key properties of zeros for L(s, ψ)L(s, χψ) in the domain Ω using various lemmas and integrative techniques.
featured image - Detailed Lemmas on Zeros of Dirichlet L-Functions in Ω
1x
Read by Dr. One voice-avatar

Listen to this story

Eigen Value Equation Population  HackerNoon profile picture
Eigen Value Equation Population

Eigen Value Equation Population

@eigenvalue

We cover research, technology, & documentation about special scalar values associated with square matrices. #EigenValue

About @eigenvalue
LEARN MORE ABOUT @EIGENVALUE'S
EXPERTISE AND PLACE ON THE INTERNET.
0-item

STORY’S CREDIBILITY

Academic Research Paper

Academic Research Paper

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Author:

(1) Yitang Zhang.

  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

4. Zeros of L(s, ψ)L(s, χψ) in Ω

In this section we prove Proposition 2.2. We henceforth assume that ψ(mod p) ∈ Ψ1. This assumption will not be repeated in the statements of Lemma 4.1-4.8.


We begin by proving some consequences of the inequalities (3.4)-(3.6).


Lemma 4.1. Let


image


image


Proof. By the Stieltjes integral we may write


image


Hence, by partial integration,


image


For G(s, ψ) an entirely analogous bound is valid. The result now follows by (3.4).


Lemma 4.2. If s ∈ Ω1, then


image


Proof. We have


image


Thus, similar to (4.2), by partial integration we obtain


image


Lemma 4.3. Let


image


We proceed to establish an approximate formula for L(s, ψ)L(s, χψ). For this purpose we first introduce a weight g(x) that will find application at various places. Let


image


with


image


We may write


image


Since


image


it follows, by changing the order of integration, that


image


Thus the function g(x) is increasing and it satisfies 0 < g(x) < 1. Further we have


image


Note that χψ is a primitive character (modDp). Write


image


so that


image


image


By (2.4) with θ = ψ and θ = χψ we have


image


This yields, by Stirling’s formula,


image


and


image


Lemma 4.4. Let


image


If s ∈ Ω3, then


image


Proof. By the residue theorem,


image


By (4.2) and (4.3),


image


By (3.5) and partial summation, the second sum on the right side above is


image


On the other hand, by the functional equation, for u = −σ − 1/2,


image


image


image


and


image


To prove (4.7) we move the contour of integration to the vertical segments



image


and to the two connecting horizontal segments


image


By a trivial bound for ω1(w), (4.5) and the residue theorem we obtain (4.7).


To prove (4.8) we move the contour of integration to the vertical segments


image


and to the two connecting horizontal segments


image


By a trivial bound for ω1(w) and (4.5) we see that the left side of (4.8) is


image


with s ∗ = 1 + α − s¯. By partial integration,


image


The estimate (4.9) follows by moving the contour of integration to the vertical segments


image


and to the two connecting horizontal segments


image


and applying (4.5) and trivial bounds for ω1(w) and the involved sum.


In order to prove Proposition 2.2, it is appropriate to deal with the function


image


By Lemma 4.2, A(s, ψ) is analytic and it has the same zeros as L(s, ψ)L(s, ψχ) in Ω1. Further, for s ∈ Ω1, we have


image


by Lemma 4.1 and 4.2. This together with Lemma 4.4 implies that


image


for s ∈ Ω3, where


image


The proof of Proposition 2.2 is reduced to proving three lemmas as follows.


Lemma 4.5. If


image


then


image


Proof. We discuss in two cases.


image


By Lemma 4.2 and trivial estimation,


image


Hence, by (4.5),


image


The result now follows by (4.10).


image


Since |B(1/2 + it, ψ)| = 1, it follows that


image


Hence, by (4.10),


image


Lemma 4.6. Suppose ρ = β + iγ is a zero of A(s, ψ) satisfying


image


Proof. It suffices to show that the function A(1/2 + iγ + w, ψ) has exactly one zero inside the circle |w| = α(1 − c ′αL), counted with multiplicity. By the Rouch´e theorem, this can be reduced to proving that


image


image


image


In either case (4.16) holds.


Lemma 4.5 and 4.6 together imply the assertions (i) and (ii) of Proposition 2.2. It is also proved that the gap between any distinct zeros of A(s, ψ) in Ω is > α(1 − c ′αL). To complete the proof of the gap assertion (iii), it now suffices to prove


image


Proof. In a way similar to the proof of Lemma 4.6, it is direct to verify that


image


We conclude this section by giving a result which is implied in the proof of Proposition 2.2.


Lemma 4.8. Assume that ρ is a zero of L(s, ψ)L(s, χψ) in Ω. Then we have


image


Proof. It follows from Lemma 4.4 that


image


image


This paper is available on arxiv under CC 4.0 license.


L O A D I N G
. . . comments & more!

About Author

Eigen Value Equation Population  HackerNoon profile picture
Eigen Value Equation Population @eigenvalue
We cover research, technology, & documentation about special scalar values associated with square matrices. #EigenValue

TOPICS

THIS ARTICLE WAS FEATURED IN...

Permanent on Arweave
Read on Terminal Reader
Read this story in a terminal
 Terminal
Read this story w/o Javascript
Read this story w/o Javascript
 Lite
Also published here
X REMOVE AD