COLS & Ramanujan's 1/π: synchrony at integers, certified tails, and a 2‑adic reading
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2025
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| _version_ | 1866901808562044928 |
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| author | POUHIER, Eric |
| author_facet | POUHIER, Eric |
| contents | <h3><strong>Description</strong></h3> <p>This expository note revisits Ramanujan’s ultra-fast series for 1/π through a COLS framework (a bank of dyadic oscillators) that makes it possible to read exact partial sums at integer times and provides a simple geometric tail bound allowing the certification of the number of decimal digits.<br>On the arithmetic side, the note gives a clear 2-adic reading of the coefficient (4k)!/(k!)⁴, with an explicit treatment of the valuations on the rational part of the terms.<br>A multibase/CRT variant faithful to 396 = 2²·3²·11 is presented in a “safe” formulation: alignment at integers without phases, and alignment at a single instant when quantized phases are introduced.<br>A short appendix provides an elementary proof that Rₖ₊₁/Rₖ > 1 for k ≥ 1, which confirms the quality of the bounds and the “digits-by-design” character of the protocol.<br>The objective is expository and certifying: to stage a classical result (Ramanujan) within a minimal and intuitive COLS framework that facilitates reproduction and verification.</p> <p><strong>Key points</strong></p> <ul> <li> <p><strong>Integer synchrony:</strong> yₖ(m)=Σₖ₌₀ᴷ Aₖ for m ∈ ℤ.</p> </li> <li> <p><strong>Geometric tail bound:</strong> Σₘ≥ᴷ₊₁ Aₘ ≤ Aₖ₊₁ / (1 – c), with uniform constant c.</p> </li> <li> <p><strong>2-adic profile:</strong> v₂((4k)!/(k!)⁴)=3 s₂(k), and v₂(rational part of Aₖ)=3 s₂(k) – 8k.</p> </li> <li> <p><strong>Multibase/CRT:</strong> safe formulation (integers) + rational phases for a single instant t*.</p> </li> <li> <p><strong>Appendix:</strong> short proof that Rₖ₊₁/Rₖ > 1 for k ≥ 1.</p> </li> </ul> <p><strong>Keywords</strong><br>Ramanujan; 1/π; COLS; dyadic synchrony; tail bound; certified digits; p-adic valuation; CRT; expository mathematics.</p> <p><strong>Related references</strong><br>– <em>COLS — Listening to Integers (V6)</em>, Zenodo, doi:<a target="_new" rel="noopener">10.5281/zenodo.17385769</a>.<br>– <em>COLS_paper_V8_FINAL</em> & <em>COLS_additional_proofs_FINAL</em>, Zenodo, doi:<a target="_new" rel="noopener">10.5281/zenodo.17383651</a>.</p> <p><em>Note – This description summarizes the results and organization of Version 7 (Proposition “Synchrony at Integers,” Tail Theorem, p-adic profiles, and monotonicity appendix).</em></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17587937 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | COLS & Ramanujan's 1/π: synchrony at integers, certified tails, and a 2‑adic reading POUHIER, Eric NumberTheory Ramanujan Pi PAdic COLS ErrorBounds CRT Hypergeometric Expository dyadic synchrony tail bound certified digits p-adic valuation <h3><strong>Description</strong></h3> <p>This expository note revisits Ramanujan’s ultra-fast series for 1/π through a COLS framework (a bank of dyadic oscillators) that makes it possible to read exact partial sums at integer times and provides a simple geometric tail bound allowing the certification of the number of decimal digits.<br>On the arithmetic side, the note gives a clear 2-adic reading of the coefficient (4k)!/(k!)⁴, with an explicit treatment of the valuations on the rational part of the terms.<br>A multibase/CRT variant faithful to 396 = 2²·3²·11 is presented in a “safe” formulation: alignment at integers without phases, and alignment at a single instant when quantized phases are introduced.<br>A short appendix provides an elementary proof that Rₖ₊₁/Rₖ > 1 for k ≥ 1, which confirms the quality of the bounds and the “digits-by-design” character of the protocol.<br>The objective is expository and certifying: to stage a classical result (Ramanujan) within a minimal and intuitive COLS framework that facilitates reproduction and verification.</p> <p><strong>Key points</strong></p> <ul> <li> <p><strong>Integer synchrony:</strong> yₖ(m)=Σₖ₌₀ᴷ Aₖ for m ∈ ℤ.</p> </li> <li> <p><strong>Geometric tail bound:</strong> Σₘ≥ᴷ₊₁ Aₘ ≤ Aₖ₊₁ / (1 – c), with uniform constant c.</p> </li> <li> <p><strong>2-adic profile:</strong> v₂((4k)!/(k!)⁴)=3 s₂(k), and v₂(rational part of Aₖ)=3 s₂(k) – 8k.</p> </li> <li> <p><strong>Multibase/CRT:</strong> safe formulation (integers) + rational phases for a single instant t*.</p> </li> <li> <p><strong>Appendix:</strong> short proof that Rₖ₊₁/Rₖ > 1 for k ≥ 1.</p> </li> </ul> <p><strong>Keywords</strong><br>Ramanujan; 1/π; COLS; dyadic synchrony; tail bound; certified digits; p-adic valuation; CRT; expository mathematics.</p> <p><strong>Related references</strong><br>– <em>COLS — Listening to Integers (V6)</em>, Zenodo, doi:<a target="_new" rel="noopener">10.5281/zenodo.17385769</a>.<br>– <em>COLS_paper_V8_FINAL</em> & <em>COLS_additional_proofs_FINAL</em>, Zenodo, doi:<a target="_new" rel="noopener">10.5281/zenodo.17383651</a>.</p> <p><em>Note – This description summarizes the results and organization of Version 7 (Proposition “Synchrony at Integers,” Tail Theorem, p-adic profiles, and monotonicity appendix).</em></p> |
| title | COLS & Ramanujan's 1/π: synchrony at integers, certified tails, and a 2‑adic reading |
| topic | NumberTheory Ramanujan Pi PAdic COLS ErrorBounds CRT Hypergeometric Expository dyadic synchrony tail bound certified digits p-adic valuation |
| url | https://doi.org/10.5281/zenodo.17587937 |