A Roughness-Explicit Chen Theorem for Goldbach Representations
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2026
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| _version_ | 1866902105728483328 |
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| author | Ratliff, Marshall |
| author_facet | Ratliff, Marshall |
| contents | <p>We prove a roughness-explicit Chen-type theorem for the binary Goldbach problem. For every fixed $\varepsilon>0$ and every sufficiently large even integer $N$, there is a representation<br>\[<br> N=p+m,\qquad p\in\mathbb P,\qquad m\in P_{\le2},\qquad P^-(m)>N^{0.2665-\varepsilon}.<br>\]<br>More precisely the same conclusion holds for every fixed $\alpha<\alpha_*$, where<br>\[<br> 0.26653256<\alpha_*<0.26653257.<br>\]<br>The proof uses a one-start Buchstab decomposition beginning at $\beta=25/224$, large-residue Goldbach distribution estimates in the Li--Lichtman framework for the base and one-prime branches, and a sign-safe Buchstab refinement of the switched Type-III upper bound for the exact three-prime contamination. The coefficient inequalities are verified by an accompanying interval-arithmetic certificate. A stronger exponent $0.2667$ is recorded only as a conditional refinement, because it requires a switched Type-III level not supplied by the quoted distribution theorem.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20377091 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Roughness-Explicit Chen Theorem for Goldbach Representations Ratliff, Marshall Sieve Theory Sieve Methods Goldbach Conjecture Rough Numbers Analytic Number Theory <p>We prove a roughness-explicit Chen-type theorem for the binary Goldbach problem. For every fixed $\varepsilon>0$ and every sufficiently large even integer $N$, there is a representation<br>\[<br> N=p+m,\qquad p\in\mathbb P,\qquad m\in P_{\le2},\qquad P^-(m)>N^{0.2665-\varepsilon}.<br>\]<br>More precisely the same conclusion holds for every fixed $\alpha<\alpha_*$, where<br>\[<br> 0.26653256<\alpha_*<0.26653257.<br>\]<br>The proof uses a one-start Buchstab decomposition beginning at $\beta=25/224$, large-residue Goldbach distribution estimates in the Li--Lichtman framework for the base and one-prime branches, and a sign-safe Buchstab refinement of the switched Type-III upper bound for the exact three-prime contamination. The coefficient inequalities are verified by an accompanying interval-arithmetic certificate. A stronger exponent $0.2667$ is recorded only as a conditional refinement, because it requires a switched Type-III level not supplied by the quoted distribution theorem.</p> |
| title | A Roughness-Explicit Chen Theorem for Goldbach Representations |
| topic | Sieve Theory Sieve Methods Goldbach Conjecture Rough Numbers Analytic Number Theory |
| url | https://doi.org/10.5281/zenodo.20377091 |