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8342 bytes, as of the version from 2026-09-17 10:33 (current). Recorded here, not run.
"""Ramanujan's class invariants G_n, with Weber's f(sqrt(-n)) -- numberdb.org/T297.
This generator fills T297 with Ramanujan's G_n and Weber's f(i*sqrt(n)),
where tau = i*sqrt(n), q = exp(-pi*sqrt(n)), and 1 <= n <= 100.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
"""
import os
import sys
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
TABLE = os.environ.get("NUMBERDB_TABLE", "T297")
DIGITS = 100
WORKING_GUARD = 256 # Widest value ball was below 2e-252 at n=99, weber.
CHECK_GUARD = 512 # Widest q-product check ball was below 3e-220 at n=1.
MAX_N = 100
PRODUCT_TERMS = 80
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
def _complex_field(digits, guard=WORKING_GUARD):
return ComplexBallField(numberdb.bits(digits, losing=guard))
def _real_field(digits, guard=WORKING_GUARD):
return RealBallField(numberdb.bits(digits, losing=guard))
def _tau(n, digits, guard=WORKING_GUARD):
field = _complex_field(digits, guard)
return field.gen(0) * field(n).sqrt()
def _weber_f_eta(n, digits, guard=WORKING_GUARD):
tau = _tau(n, digits, guard)
value = tau.modular_eta() ** 2 / ((tau / 2).modular_eta() * (2 * tau).modular_eta())
if not value.imag().contains_zero():
raise ArithmeticError("Weber f(i*sqrt(%s)) has nonzero imaginary part: %s" % (n, value))
real = value.real()
if not real > 0:
raise ArithmeticError("Weber f(i*sqrt(%s)) is not positive: %s" % (n, real))
return real
def _ramanujan_g(n, digits, guard=WORKING_GUARD):
field = _real_field(digits, guard)
return field(_weber_f_eta(n, digits, guard)) / field(2) ** (QQ(1) / QQ(4))
def _weber_f_product(n, digits, terms=PRODUCT_TERMS, guard=CHECK_GUARD):
"""Weber f from the q-product, with a positive tail bound."""
field = _real_field(digits, guard)
n = field(n)
q = (-field.pi() * n.sqrt()).exp()
product = field(1)
for m in range(1, terms + 1):
product *= 1 + q ** (2 * m - 1)
leading = (field.pi() * n.sqrt() / 24).exp()
partial = leading * product
tail_sum = q ** (2 * terms + 1) / (1 - q ** 2)
error = partial * (tail_sum.exp() - 1)
return partial.add_error(error)
def _lambda_parameter(n, digits, guard=CHECK_GUARD):
value = _tau(n, digits, guard).modular_lambda()
if not value.imag().contains_zero():
raise ArithmeticError("lambda(i*sqrt(%s)) has nonzero imaginary part: %s" % (n, value))
real = value.real()
if not (real > 0 and real < 1):
raise ArithmeticError("lambda(i*sqrt(%s)) is not contained in (0, 1): %s" % (n, real))
return real
def _j_invariant(n, digits, guard=CHECK_GUARD):
value = _tau(n, digits, guard).modular_j()
if not value.imag().contains_zero():
raise ArithmeticError("j(i*sqrt(%s)) has nonzero imaginary part: %s" % (n, value))
return value.real()
class RamanujanClassInvariantsG(numberdb.Generator):
"""Generator for T297, Ramanujan's G_n and Weber's f(i*sqrt(n))."""
table = TABLE
parameters = ("n", "normalisation")
type = "R"
digits = DIGITS
rigour = "proven"
def enumerate(self, max_n=MAX_N):
for n in range(1, max_n + 1):
yield {"n": str(n), "normalisation": "ramanujan"}
yield {"n": str(n), "normalisation": "weber"}
def value(self, params, digits):
n = ZZ(params["n"])
normalisation = params["normalisation"]
if normalisation == "ramanujan":
if n == 1:
return ZZ(1)
return _ramanujan_g(n, digits)
if normalisation == "weber":
return _weber_f_eta(n, digits)
raise ValueError("unknown normalisation %r" % (normalisation,))
def _overlaps_zero(value):
return value.contains_zero()
def run_integrity_checks():
field = _real_field(DIGITS, CHECK_GUARD)
worst_value_radius = field(0)
worst_value_at = None
worst_check_radius = field(0)
worst_check_at = None
for n in range(1, MAX_N + 1):
f = _weber_f_eta(ZZ(n), DIGITS, CHECK_GUARD)
g = field(f) / field(2) ** (QQ(1) / QQ(4))
product_f = _weber_f_product(ZZ(n), DIGITS)
if not _overlaps_zero(product_f - f):
raise ArithmeticError("q-product check failed at n=%d: %s vs %s" % (n, product_f, f))
m = _lambda_parameter(ZZ(n), DIGITS, CHECK_GUARD)
singular_relation = g ** (-24) - 4 * m * (1 - m)
if not _overlaps_zero(singular_relation):
raise ArithmeticError("singular-modulus relation failed at n=%d: %s" % (
n, singular_relation))
reciprocal = _ramanujan_g(QQ(1) / QQ(n), DIGITS, CHECK_GUARD)
if not _overlaps_zero(reciprocal - g):
raise ArithmeticError("reciprocal relation failed at n=%d: %s vs %s" % (
n, reciprocal, g))
j_formula = (f ** 24 - 16) ** 3 / f ** 24
j_value = _j_invariant(ZZ(n), DIGITS, CHECK_GUARD)
if not _overlaps_zero(j_formula - j_value):
raise ArithmeticError("j-Weber relation failed at n=%d: %s vs %s" % (
n, j_formula, j_value))
for normalisation, value in (("ramanujan", g), ("weber", f)):
radius = field(value.rad())
if radius > worst_value_radius:
worst_value_radius = radius
worst_value_at = (n, normalisation)
radius = field(product_f.rad())
if radius > worst_check_radius:
worst_check_radius = radius
worst_check_at = (n, "q-product")
print("integrity checks passed for n=1..%d" % MAX_N)
print("widest value ball radius: %s at n=%s, %s" % (
worst_value_radius, worst_value_at[0], worst_value_at[1]))
print("widest check ball radius: %s at n=%s, %s" % (
worst_check_radius, worst_check_at[0], worst_check_at[1]))
def fill_draft_once(generator, message):
"""Fill a fresh prose draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator, os.environ.get("NUMBERDB_ASSISTED_BY", "")),
upsert=False,
run=run,
rigour=generator.rigour,
)
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message, rigour=generator.rigour)
return answer
if __name__ == "__main__":
_key_from_stdin()
generator = RamanujanClassInvariantsG()
run_integrity_checks()
if "--publish" in sys.argv or os.environ.get("NUMBERDB_PUBLISH") == "1":
print(fill_draft_once(
generator,
message="Ramanujan class invariants G_n and Weber f(i*sqrt(n)) for n=1..100"))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)