lucidia sync: providers registry, db, main updates

codex/mirror/number_mirror_mu.py

@@ -1,65 +0,0 @@
-"""
-number_mirror_mu.py
-
-This module implements a simple Möbius mirror demonstration.
-It defines functions to compute the Möbius function µ(n), split
-positive and negative values, compute the Mertens function, and
-verify the Dirichlet generating identity.
-
-Functions:
-    mobius(n) -> int
-    mirror_split_mu(N) -> (pos_indices, neg_indices)
-    mertens(N) -> list[int]
-    dirichlet_sum(s, N) -> complex
-"""
-
-import cmath
-
-def mobius(n: int) -> int:
-    """Compute the Möbius function µ(n)."""
-    if n == 1:
-        return 1
-    primes = {}
-    i = 2
-    m = n
-    while i * i <= m:
-        while m % i == 0:
-            primes[i] = primes.get(i, 0) + 1
-            m //= i
-        i += 1
-    if m > 1:
-        primes[m] = primes.get(m, 0) + 1
-    for exp in primes.values():
-        if exp > 1:
-            return 0
-    return -1 if len(primes) % 2 else 1
-
-def mirror_split_mu(N: int):
-    """Return indices where µ(n) = +1 and µ(n) = -1 up to N."""
-    pos = []
-    neg = []
-    for n in range(1, N + 1):
-        mu = mobius(n)
-        if mu == 1:
-            pos.append(n)
-        elif mu == -1:
-            neg.append(n)
-    return pos, neg
-
-def mertens(N: int):
-    """Compute the Mertens function M(x) for x = 1..N."""
-    total = 0
-    M = []
-    for n in range(1, N + 1):
-        total += mobius(n)
-        M.append(total)
-    return M
-
-def dirichlet_sum(s: complex, N: int):
-    """Compute the partial Dirichlet sum \u2211_{n=1..N} µ(n)/n^s."""
-    total = 0+0j
-    for n in range(1, N + 1):
-        mu = mobius(n)
-        if mu != 0:
-            total += mu / (n ** s)
-    return total