🎯 Function Selection
⚙️ Parameters
Theoretical Background
The function Ω(n) counts prime factors with multiplicity. For n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ:
Ω(n) = a₁ + a₂ + ... + aₖ
Our research shows that the finite-size deviation from uniform distribution follows:
|S(x)|/x ~ C_m × (log x)^(cos(2π/m) - 1)
📊 Decay Analysis
Finite-Size Law
For m = 3, the first Fourier coefficient decays as:
|S(x)|/x ~ 1.708 × (log x)^(-3/2)
This matches the Selberg-Delange/Halász prediction with cos(2π/3) - 1 = -3/2.
Note: The constant C₃ = 1.708 is computed using natural logarithms (ln), not log₁₀.
〰️ Fourier Analysis
Fourier Decomposition
All deviations from uniformity are captured by the single complex sum:
S(x) = Σ_{n≤x} e^(2πi·Ω(n)/m)
⚖️ Weighted Analysis
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