# Ancillary files for "The exact second-order term in Ramanujan's bound for sigma(n)/n" (Jacob Crim).
# Requirements: Python 3.10+, python-flint (Arb ball arithmetic), sympy, mpmath; gcc + libquadmath for the sieve.
files:
  symbolic.py: >-
    Exact sympy checks: cancellation of the s^0 and s^1 terms, U1(0)=8, M1(0) independent of v.
    Writes sym.pkl and lemma.pkl (needed by engine.py and lemma_shell2.py). Run this first.
  lemma_shell2.py: "Proof of Lemma (shells): v in [0.47, 0.55] for all real xi >= 1e9."
  cconst.py: "C = sup_{0<y<=1423} (y - theta(y))/sqrt(y) = 2.05282..."
  engine.py: "Enclosure of t^2 E on a closed t-cell (Appendix A of the paper). SHA-256 in hashes.log."
  cover.py: "Covering of t in [log 1e9, 300]. Usage: python3 cover.py <t_start> <t_end>."
  cells.csv: >-
    Output of the covering, 30,798 cells, one row per cell: ta, tb (the t-interval, full precision,
    contiguous from log 1e9 to 300); Elo, Ehi (bounds for t^2 E); Llo, Lhi (bounds for
    t*(c_sigma/e^gamma - (2 - 2 sqrt2 + W))); cup (upper bound for c_sigma/e^gamma); app (lower bound
    for c_sigma/e^gamma - (2 - 2 sqrt2 + W)). Bounds are rounded outward to 8 significant digits.
    cover.py writes the same data unrounded as JSON lines.
  tail.py: "Point evaluations of the non-analytic terms at t = 300 ... 4800 (consistency check)."
  tail2.py: "Bounds U1+M1+U6 on 200 subintervals of s in [0, 1/300] (the region t >= 300)."
  sieve_build.py: "Writes and compiles the C segmented sieve to 1e10 (source embedded)."
  validate.py: "Compares every group of enclosures with exact values at CA numbers in [1e9, 1e10]."
  oddcheck.py: "Checks the odd-integer corollary bound (t + log t + 0.37)/sqrt(xi)."
  validation_output.log: "Output of validate.py at 101 checkpoints (no violations)."
  oddcheck_output.log: "Output of oddcheck.py at 101 checkpoints (no violations)."
  hashes.log: "SHA-256 of engine.py, sym.pkl and lemma.pkl used for cells.csv."
reproduce:
  - "python3 symbolic.py"
  - "python3 lemma_shell2.py"
  - "python3 cover.py 20.72326583694641 300   # ~30 min on 2 cores; compare with cells.csv"
  - "python3 tail2.py"
  - "python3 sieve_build.py && mkdir -p ../sieve && ./sieve 10000000000 401 1e9 1e10 > ../sieve/cp_1e9_1e10.txt"
  - "put the first ~1942 zeta-zero ordinates, one per line, in ../zeros.txt (e.g. mpmath.zetazero)"
  - "python3 validate.py 4 ; python3 oddcheck.py"