The Amparo LC Conjecture: Finiteness of the Correlation Energy of the Liouville Function
Year: 2026
Type: preprint
Abstract: We formulate a conjecture on the convergence of the correlation energy series of the Liouville function. For every fixed integer h ≥ 1, let S_h(x) = Σ_{n≤x} λ(n)λ(n+h) and Z_h(3) = Σ_{n=1}^∞ S_h(n)²/n³. The Amparo LC Conjecture states that Z_h(3) converges to a finite constant. We prove that this conjecture is strictly equivalent to Chowla's conjecture (1965) for k=2 (Universal Reduction Theorem). We establish an Inertia Lemma: if Z_h(3) < ∞, then the exceptional set E_h(ε) = {n : |S_h(n)/n| ≥ ε} is finite for every ε > 0. Computational evidence with N = 200,000 verifies convergence to 1.3765, with 99.97% of the series concentrated in n ≤ 10,000. The convergence abscissa satisfies σ_c ≤ 0.5. The conjecture reduces the 61-year-old Chowla problem to the finiteness of a single scalar.
Author: Durango López, Germán Alonso
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162207302
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