Python verification scripts for the paper
"Thinned Wallis-type prime products in residue classes modulo 2^m"

Main file:
    tw_python_verification_fft.py

Dependencies:
    Python 3
    mpmath

Purpose:
    The script evaluates the character-sum formula for log K(2^m,S).
    It uses arbitrary-precision arithmetic and computes Dirichlet L-values
    via Hurwitz zeta functions and the digamma formula at s=1.  For speed,
    it evaluates all characters modulo 2^m by a radix-2 finite Fourier
    transform on (Z/2^m Z)^* = {+-1} x <5>.

Examples:
    python tw_python_verification_fft.py --m 4 --S 1 15 --nmax 40 --dps 80
    python tw_python_verification_fft.py --m 4 --S 3 11 --nmax 64 --dps 90
    python tw_python_verification_fft.py --m 10 --S 1 1023 --nmax 20 --dps 45

Stability check:
    python tw_python_verification_fft.py --m 4 --S 1 15 --stability --nvals 16 24 32 40 --dps 80

Direct product check, used only for slow consistency checks:
    python tw_python_verification_fft.py --direct --m 4 --S 3 11 --x 100000000 --dps 40

Notes:
    The direct products converge much more slowly than the character-sum
    formula.  For the unbalanced example S={3,11} mod 16 the normalized
    value at x=10^8 is already close to the character-sum value.

License suggestion:
    If these scripts are published with the paper, state an explicit license,
    for example MIT, BSD-2-Clause, or GPL-3.0.  If code is adapted from
    Languasco's GPL PARI/GP programs, the resulting code should respect the
    GPL.  The present script is an independent Python implementation of the
    character formula and does not copy PARI/GP source code.