Finite-Size Equidistribution of Omega(n) Modulo m: Theory and Computation

We study the finite-size distribution of the additive prime factor count Omega(n) modulo m. Using Fourier analysis and the Selberg-Delange method, we derive explicit decay laws for residue class imbalances and verify predictions computationally up to x = 10^8. The flagship case m=3 exhibits |S(x)|/x ~ 1.708(log x)^{-3/2}, confirmed to 6 decimal places. We extend to m = 4,5,6 and demonstrate universality across omega(n) and Omega(n). Short-interval analysis reveals a threshold H ≳ x^{0.6} for the decay law to manifest locally.

 

 

## Methods

The computational approach combines four key techniques:

1. **SPF Sieve**: O(N log log N) preprocessing enables O(log n) factorization per number

2. **Euler Product**: Theoretical constants via truncated product over primes (10^6 primes ≈ 6 decimal accuracy)

3. **Dyadic Shell Regression**: Power-law fitting on [2^k, 2^{k+1}] intervals reduces autocorrelation

4. **Bootstrap**: 1000-sample resampling (seed=42) for uncertainty quantification

**Reference**: Paper Section 3 (Computational Methods)

## Key Theoretical Results

### The Decay Law

For any modulus m, the Fourier coefficient decays as:

```

|S(x)|/x ~ C_m (log x)^{cos(2π/m) - 1}

```

where `S(x) = sum_{n ≤ x} omega^{Omega(n)}` with `omega = exp(2πi/m)`.

### Theoretical Constants (Paper Table 1)

**Empirical estimates**: Dyadic shell regression on x ∈ [2^{10}, 2^{26}], bootstrap 95% CI

### Short Intervals

The decay law also holds in short intervals [x, x+H] provided:

- H ≳ x^{0.6} (threshold for law to manifest)

- Optimal window: H ≈ x^{0.8} (balances noise vs. bias)

**Reference**: Paper Section 5 (Short Intervals)

## Reproducibility

All computational results in the paper are fully reproducible:

**Hardware**: Intel Core i9-12900K, 64GB RAM (consumer-grade)

**Software**: Python 3.11.5, NumPy 1.24.3, Matplotlib 3.7.1, SciPy 1.10.1

**Seed**: 42 (for all bootstrap sampling)

**Dyadic ranges**: [2^k, 2^{k+1}] for k = 10, 11, ..., 26

**Runtime**: ~45 min for m=3 at N=10^8 (single-threaded)

Results are stored in `data/omega_constants_final.json` and can be verified against paper values.