"""euler_law.py -- verifies the exact Euler-product / L-value law
   W(k/6; 2g,3g) = -chi3(g k) * 3 L(1,chi3) * prod_{p|g} (1 - chi3(p)/p) / (g k),
   with 3 L(1,chi3) = pi/sqrt(3), for (g,6)=(k,6)=1.   Agreement ~1e-15.
"""
from wcollapse import W_collapsed
from math import gcd, pi, sqrt

def chi3(n):
    n %= 3
    return 0 if n == 0 else (1 if n == 1 else -1)

def factor(n):
    f, p, m = [], 2, n
    while p * p <= m:
        if m % p == 0:
            f.append(p)
            while m % p == 0: m //= p
        p += 1
    if m > 1: f.append(m)
    return f

def predicted(g, k):
    pr = 1.0
    for p in factor(g): pr *= (1 - chi3(p) / p)
    return -chi3(g * k) * (pi / sqrt(3)) * pr / (g * k)

if __name__ == "__main__":
    mx = 0.0
    for g, k in [(1,1),(5,1),(7,1),(11,5),(13,7),(35,7),(91,13),(143,11),
                 (187,25),(209,35),(221,1),(247,49)]:
        if gcd(g, 6) != 1 or gcd(k, 6) != 1: continue
        w = W_collapsed(2*g, 3*g, k, 6); p = predicted(g, k)
        mx = max(mx, abs(w - p))
        print(f"g={g:4d} k={k:3d}: W={w:+.14f} predicted={p:+.14f} diff={abs(w-p):.1e}")
    print("max diff:", mx)