Python scripts for figures in:

"On Landau–Siegel zeros and heights of singular moduli" arXiv:1911.07215

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Description

Fixed files

• generators/listgen.py: Generator of lists (negFDs.txt and hjtds.txt simultaneously, described below).

• generators/figure_1.py: Generator of the image (described below)

• params.txt: Tells generators the upper limit and precision for values generated for lists (default: negative fundamental discriminants up to -10^6, precision up to 10^{-16})

• lpl1chiD/*: Additional scripts to calculate and plot L'(1, chi_D)/L(1, chi_D) (cf. lpl1chiD/README.md)

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Generated files

Images

• imgs/figure_1eN.png: Graph of ht(j(tau_D))/3log(|D|) for D negative fundamental discriminants with |D| <= 10^N, where:
  • ht = absolute logarithmic Weil height,
  • j = Klein's classical j-invariant function,
  • tau_D = sqrt(D)/2 if D == 0 (mod 4), (-1+sqrt(D))/2 if D == 1 (mod 4),

Lists

• lists/negFDs_1eN.txt: Negative fundamental discriminants for D > -10^N

• lists/hjtds_1eN.txt: List of the values of ht(j(tau_D))/3log(|D|) for D > -10^N

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How to run

Linux

1. Run gen.sh (can be halted and resumed at any point)

Not-linux

1. Run generators/listgen.py (can be halted and resumed at any point)
2. Run generators/figure_1.py

In both cases, lists are stored in ~/lists and images are stored in ~/imgs.

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Dependencies

All the Python scripts were written for Python 2.7. Apart from standard Python libraries, mpmath and Matplotlib are necessary to run the scripts.

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Time estimate

As a first version, the algorithms were not exactly written with efficiency in mind. For example, a laptop with the specs

• Processor: Intel(R) Core(TM) i3-5005U CPU @ 2.00GHz,
• Memory: 4GiB, clock 1600MHz (0.6ns),
• HDD: WDC WD5000LPCX-6 500GB,

running listgen.py until 10^6 takes approximately 30 hours, whilst running it until 10^7 takes a good portion of an entire month.