Landau-Siegel Zero Tester

This is part of an undergraduate research project at University of Toronto, Math Department that aims to verify the non-existence of Landau-Siegel zero in the region $\beta_1 > 1-\frac{c}{\log(q)}$ for primitive quadratic characters mod $q$, with $0<q<q_{max}$. This project was funded by NSERC and UTEA Undergraduate Student Research Awards. This research is also supported by Compute Canada (alliancecan.ca), where most of our computation was performed.

Table of Contents

1. background
2. Requirements
3. Usage
4. Directory Structure

Background

Let $\chi$ be a Dirichlet character mod $q$, $$L(s,\chi) := \sum_{n=1}^\infty \frac{\chi(n)}{n^s}, \quad \Re(s) > 0,$$ be the associated Dirichlet $L$-function, and $s = \sigma + it$ be a complex number.

Landau proved that there is a constant $c > 0$ such that for all $q$, $\prod_\chi L(s,\chi)$ has at most one zero in the region

$$ \sigma > 1 - \frac{c}{\log(q(|t|+4))} $$

where $\chi$ ranges over all Dirichlet characters of modulus $q$.
If such a zero exists, then it is necessarily real and the associated character $\chi$ is quadratic.

Such a zero, if it exists, is called a Landau–Siegel zero or an exceptional zero.

Running this program on Compute Canada's Nibi clusters, we computationally verified that Landau–Siegel zeros do not exist for any primitive quadratic character of modulus $q \le 10^{9}$. In particular, we proved the following theorem.

Theorem 1.
Let $\chi$ be a primitive quadratic character of modulus $q \leq 10^{9}$. Then $L(s,\chi)$ has no real zeros $\beta_1$ in the region

$$ \beta_1 > 1 - \frac{c}{\log q} $$

with $c = \tfrac{1}{10}$.

Requirements

To build and run the computational verification code, you will need the following system dependencies:

• FLINT (Fast Library for Number Theory) ≥ 2.9
• CMake ≥ 3.20 (for building the project)
• MPI (Message Passing Interface, e.g. OpenMPI or MPICH) for parallel execution

Make sure these are installed and available in your system path.

Usage

After installing the requirements and building the project, you can run the computations as follows.

1. Precompute chi

Run 'misc/precompute_kronecker.c' to generate chi'txt. After the file is generated, move it to the /input dirctory.

2. Choose the version

Edit src/CMakeLists.txt to choose the version of main program to run (either main.c, main_by_file.c, or main_test_ind.c).

3. Build the project

From the repository root:

mkdir build
cd build
cmake ..
cmake --build .

4. Run with MPI

From the repository root use the following mpi command to run, change 10 to desired number of MPI processes.

mpirun -np 10 ./build/src/mpiTest

5. Using main_by_file.c

If one wants to run the verification on a list of $q$ in a file, name the file to input.txt and store it in /input dirctory.

Directory Structure

All source code is stored in the directory /src. All input files are stored in /input dirctory. /test dirctory contains all unit tests for this project and /misc conatins all other helping code and scripts.

By Rick Lu, Asif Zaman, and Haonan Zhao