Parity Bias in Square-Free Wheel Sieves: An Exact Liouville Identity and the ζ(1/2) Amplitude Constant
Year: 2026
Type: preprint
Abstract: Mathematical Summary & Theoretical Scope We study the parity distribution of $\Omega(n)$ (the number of prime factors counted with multiplicity) across the integers coprime to a square-free modulus $P$. The study establishes three complementary levels of results: Exact Unconditional Liouville Identity: We prove that for any square-free integer $P \ge 2$ and every real $X \ge 2$, $$\sum_{\substack{n \le X \\ \gcd(n,P)=1}} \lambda(n) = \sum_{d \mid P} M(X/d)$$ where $\lambda$ is the Liouville function and $M$ is its summatory function. Consequently, the deviation from equidistribution is given exactly by $\mathrm{dev}(P,X) = -(2N_P(X))^{-1}\sum_{d \mid P} M(X/d)$, with no error term. Conditional Rate under the Riemann Hypothesis (RH): Assuming RH, we establish that $\mathrm{dev}(P,X) \ll B(P) X^{-1/2} \exp(C \log X / \log\log X)$, proving that the bias decays as $X^{-1/2+o(1)}$. The Multiplicative Amplitude Law: We show that the governing amplitude is given by the exact constant $$C_\zeta(P) = \frac{1}{2\vert{}\zeta(1/2)\vert{}} \prod_{p \mid P} \frac{1 + p^{-1/2}}{1 - p^{-1}}$$ which factors multiplicatively over the primes dividing $P$. The normalized bias $X^{1/2}\mathrm{dev}(P,X)$ fluctuates around this mean with explicit phase modulations governed by the non-trivial zeros $\rho = 1/2 + i\gamma$ of the Riemann zeta function. Structure of the Replication Package The repository contains the companion Jupyter notebooks that orchestrate the end-to-end execution, paired with their respective cryptographic ZIP archives: 1. Formal Lean 4 Verification Suite Interactive Notebook: Lean4_Parity_Bias_SFWS.ipynb Source Archive: Lean4_Parity_Bias_SFWS.zip Cryptographic Hash (SHA-256): 089d9376c1fd318b7d1235645254f2df3fa857419b194f4907279c8eeca99cb0 Verification Scope: 31 self-contained modules in Lean 4 (v4.35.0) with Mathlib4, certifying 213 theorems/lemmas and 74 definitions with 0 sorry gaps and 0 global axiom declarations. Certifies the algebraic skeleton: modular reductions, the exact Liouville identities, primorial towers, deterministic sign alternations $\Delta_r = (-1)^{r+1}$, and the sifted Euler product factorization. 2. Empirical Computational & Statistical Pipeline Interactive Notebook: Exp_Parity_Bias_SFWS.ipynb Source Archive: Exp_Parity_Bias_SFWS.zip Cryptographic Hash (SHA-256): 626927ebeb586a8df3c0e95f0dfacd75387465b86e7a68f066800168e07dfae3 Experimental Scope: Compiled Eratosthenes–Legendre sieve in CPU and CuPy/CUDA bit-packed kernels up to $X = 10^{10}$. Out-of-sample 91-point spectral reconstruction on $[10^9, 10^{10}]$ using the first 200 non-trivial Riemann zeros ($r = 0.9865$, capturing $97.3\%$ of the variance with predicted theoretical amplitudes). Cross-primorial longitudinal audits across 6 primorials ($P \in \{6, 30, 210, 2310, 30030, 510510\}$) over 5 scales (grand mean ratio $1.010 \pm 0.009$). High-energy audit of Open Problem (O5) across $10^4$ non-trivial zeros ($T \approx 9877.78$) and Monte Carlo evaluation ($500{,}000$ samples) of the limiting Rubinstein–Sarnak measure $\nu_6$ on $\mathbb{T}^{200}$. Usage & System Requirements Lean 4: Requires elan and lake with toolchain synchronized to Mathlib4 v4.35.0. Run lake exe cache get! and lake build. Python Pipeline: Python 3.11+, NumPy, SciPy, Numba, mpmath (50-digit precision), CuPy (optional, for CUDA GPU acceleration), and Matplotlib. Submitted to Acta Mathematica Sinica, English Series.
Author: Peinador Sala, José Ignacio
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225610087