AI 中文总结
研究黎曼ξ函数系数连续托普利兹子式，通过结合复鞍点分析、\(q\)-帕斯卡扩张半群等方法，证明对于\(r\geq2\)和\(k\geq10^{18}r^3\)，\(D_{r,k}>0\)，给出显式三次尾部尺度，未涉及黎曼假设互补区域。
AI 中文摘要
设\((a_k)\)为归一化黎曼ξ函数的正系数序列，\(D_{r,k}\)表示其连续托普利兹子式。黎曼假设等价于\((a_k)\)是无穷阶的波利亚频率序列，进而等价于所有托普利兹子式非负。我们证明对于每个\(r\geq2\)和\(k\geq10^{18}r^3\)，\(D_{r,k}>0\)。这给出了一个在\(r\)上一致的显式三次尾部尺度，与卡特科娃的固定阶渐近正性不同。证明未使用黎曼ζ函数的数值验证零点。它结合了矩变换的经认证的复鞍点分析、同时控制每个次数的精确\(q\)-帕斯卡扩张半群以及非线性余项的加权巴拿赫代数主项，并通过保惯性论证封闭。所有解析常数在阿尔布球算术中有向舍入下得到认证，所有代数恒等式在精确有理算术中得到验证。辅助文件重现了每个证书。结果仅涉及\(k\)远大于\(r^3\)的尾部区域，对黎曼假设所涉及的互补区域没有进展。
英文摘要
Let (a_k) be the positive coefficient sequence of the normalized Riemann xi-function, and let D_{r,k} denote its consecutive Toeplitz minors. The Riemann Hypothesis is equivalent to (a_k) being a Polya frequency sequence of infinite order, and hence to nonnegativity of all Toeplitz minors. We prove that D_{r,k} > 0 for every r >= 2 and k >= 10^18 r^3. This gives an explicit cubic tail scale uniform in r, in contrast with Katkova's fixed-order asymptotic positivity. The proof does not use numerically verified zeros of the Riemann zeta-function. It combines a certified complex saddle-point analysis of the moment transform, an exact q-Pascal dilation semigroup controlling every degree simultaneously, and a weighted Banach-algebra majorant for the nonlinear remainder, closed by an inertia-preservation argument. All analytic constants are certified with directed rounding in Arb ball arithmetic, and all algebraic identities are verified in exact rational arithmetic. The ancillary files reproduce every certificate. The result concerns only the tail regime k much larger than r^3 and makes no progress on the Riemann Hypothesis, which concerns the complementary region.
Comments15 pages; ancillary files provide reproducible Arb ball-arithmetic certificates, exact-arithmetic verification code, and 36 automated tests
