mirror of
https://github.com/blackboxprogramming/lucidia.git
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Alexa Amundson
855585cb0e
Synced from BlackRoad-OS-Inc/blackroad-operator/orgs/personal/lucidia BlackRoad OS — Pave Tomorrow. RoadChain-SHA2048: fe729062952871e7 RoadChain-Identity: alexa@sovereign RoadChain-Full: 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
66 lines
1.6 KiB
Python
66 lines
1.6 KiB
Python
"""
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number_mirror_mu.py
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This module implements a simple Möbius mirror demonstration.
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It defines functions to compute the Möbius function µ(n), split
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positive and negative values, compute the Mertens function, and
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verify the Dirichlet generating identity.
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Functions:
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mobius(n) -> int
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mirror_split_mu(N) -> (pos_indices, neg_indices)
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mertens(N) -> list[int]
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dirichlet_sum(s, N) -> complex
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"""
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import cmath
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def mobius(n: int) -> int:
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"""Compute the Möbius function µ(n)."""
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if n == 1:
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return 1
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primes = {}
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i = 2
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m = n
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while i * i <= m:
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while m % i == 0:
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primes[i] = primes.get(i, 0) + 1
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m //= i
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i += 1
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if m > 1:
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primes[m] = primes.get(m, 0) + 1
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for exp in primes.values():
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if exp > 1:
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return 0
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return -1 if len(primes) % 2 else 1
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def mirror_split_mu(N: int):
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"""Return indices where µ(n) = +1 and µ(n) = -1 up to N."""
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pos = []
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neg = []
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for n in range(1, N + 1):
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mu = mobius(n)
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if mu == 1:
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pos.append(n)
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elif mu == -1:
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neg.append(n)
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return pos, neg
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def mertens(N: int):
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"""Compute the Mertens function M(x) for x = 1..N."""
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total = 0
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M = []
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for n in range(1, N + 1):
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total += mobius(n)
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M.append(total)
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return M
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def dirichlet_sum(s: complex, N: int):
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"""Compute the partial Dirichlet sum \u2211_{n=1..N} µ(n)/n^s."""
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total = 0+0j
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for n in range(1, N + 1):
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mu = mobius(n)
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if mu != 0:
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total += mu / (n ** s)
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return total
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