COUNTING ZEROS OF DIRICHLET L-FUNCTIONS

Counting zeros of Dirichlet L-Functions

Michael A. Bennett, Greg Martin, Kevin O'Bryant & Andrew Rechnitzer

Manuscript

The manuscript has been submitted and and also appears on the arxiv.

Supplementary documents

A short supplementary document which gives some more details on how various inequalities in Proposition 3.2, Lemma 3.4 and Section 6.1 were verified using interval analysis.

Mathematica Packages

The results in the paper rely on substantial calculations in Mathematica. To facilitate those computations Kevin O'Bryant created a number of Mathematica pacakges. These may be of interest more generally so we give links to their repositories.

DirichletCharacters`: Mathematica package for handling Dirichlet Characters and accessing their zeros. It uses the same notation as the online database LMFDB.
QueueStack`: Mathematica package implementing queues and stacks.
IntervalTools`: Mathematica package for doing interval analysis computations, and proving rigorous inequalities.
Versions of the packages as of May 8, 2020 (for archival's sake): DirichletCharacters` (and the associated dataset), IntervalTools`, QueueStack`

Code used in computations

Inequalities in Section 1

Mathematica notebook (PDF) that verifies the various claims in the introduction.

Proposition 3.2 - \( g(a,T) \) inequality

Mathematica notebook (PDF) that computes a rigorous bound on \( T g(a,T) \), and

Lemma 3.4 - \( E(a,d,T) \) inequality

Mathematica notebook (PDF) proving that \(E(a,d,T)\) is a increasing positive function, and

Section 6.1 - Large values of \(\ell\)

Mathematica notebook (PDF) that proves the main theorem for \(\ell \geq 27.02 \)

Section 6.2 - Middle values of \(\ell\)

Mathematica notebook (PDF) that proves the main theorem for \(5.98 \leq \ell \leq 28 \)

Section 6.3 and Lemma 6.1 - Small values of \(T\) and \(\ell\)

A gzip'd text file of \(L\)-function zeros for conductors \(3 \leq q \leq 934\) and heights corresponding to \(\ell \leq 6\). Note that there is no \(L\)-function for conductor 934.
The number of zeros was computed rigorously using arblib. The zeros themselves were computed quickly but non-rigorously using pari-gp and then confirmed rigorously using arblib. Some simple python code was used to stitch the other code together.
- Python wrapper - outer that runs code for all required conductor.
- Python wrapper - inner that runs code for specific \(L\)-functions.
- c++/arblib code to compute the number of zeros of a given \(L\)-function for a given height.
- Pari-gp code for computing zeros of real and imaginary characters.
- c++/arblib code to verift zeros of a given \(L\)-function for a given height.
A Mathematica notebook to convert data from the above process into a Mathematica-friendly dataset. The resulting files are available in a gzip'd-tar file.
Mathematica notebook (PDF) that processes the \(L\)-function zeros and proves the main theorem for \(0 \leq \ell \leq 6\).