对数积分渐近展开的加权素数幂截断：性质与应用
Weighted Prime Powers Truncation of the Asymptotic Expansion for the Logarithmic Integral: Properties and Applications
AI总结：
该研究提出连续几何变分优化范式，构建无余项参数算子曲面重构素数分布，通过临界线变分极小值性质推导非平凡零点约束，为黎曼假设提供变分证据，且未声称证明该假设。
摘要：
我们提出一种新颖的连续几何范式，通过对光滑参数曲面进行变分优化来重构素数的分布规律。通过放宽传统级数截断的严格整数约束，我们构建了无余项的参数算子曲面$\u2009li(n; τ)$，该曲面在连续坐标域$(1, \infty) \times \mathbb{R}^+$上均匀映射。我们借助不变七次混合多项式$ϕ(x) = 35x^4 - 84x^5 + 70x^6 - 20x^7$证明了全局$C^3$多变量正则性，完全消除了标准分段分数插值中存在的人为阶跃不连续性和导数扭结。通过施加严格的素数截断恒等式$\u2009li(n; τ_π(n)) = π(n)$，我们证明了素数计数函数的离散、非光滑算术谱可以唯一映射到连续无间断的截断$τ_π(n)$上。\n 通过引入Littlewood无条件振荡界，并将Riemann–von Mangoldt公式应用于截断函数$τ(n)$，我们证明了临界线是唯一的绝对全局变分极小值。利用这一优化约束，我们通过反证法证明：任何假设的离线零点簇（$β_0 \neq 1/2$）在数学上都会导致截断参数无法在无穷尺度范围内始终保持正实数值。为保证流形的实值存在性，所有非平凡零点都被约束在该绝对全局极小值上，迫使$β$恒等于$1/2$，为黎曼假设（RH）的证明提供了变分证据。*注：本手稿并未声称证明了黎曼假设，相关细节在结论部分说明。
英文摘要：
We introduce a novel, continuous geometric paradigm to reframe the distribution of prime numbers through the variational optimization of a smooth parameter surface. By relaxing the rigid integer constraints of traditional series truncations, we construct a remainder-free parametric operator surface $\operatorname{li}(n; τ)$ mapped uniformly across the continuous coordinate domain $(1, \infty) \times \mathbb{R}^+$. Global $C^3$ multi-variable regularity is established via an invariant septic blending polynomial $ϕ(x) = 35x^4 - 84x^5 + 70x^6 - 20x^7$, completely neutralizing the artificial step-discontinuities and derivative kinks of standard piecewise fractional interpolations. By enforcing the strict prime truncation identity $\operatorname{li}(n; τ_π(n)) = π(n)$, we demonstrate that the discrete, non-smooth arithmetic spectrum of the prime-counting function can be mapped uniquely onto an unbroken, continuous truncation $τ_π(n)$. By transferring Littlewood's unconditional oscillation bounds and applying the Riemann--von Mangoldt formula to the truncation functions $τ(n)$. We establishing the critical line as the unique, absolute global variational minimum. Exploiting this optimization constraint, we show via proof by contradiction that any hypothetical off-line zero cluster ($β_0 \neq 1/2$) would mathematically forbid the truncation parameter from remaining positive and real-valued across the infinite scale horizon. To preserve the real-valued existence of the manifold, all non-trivial zeroes are shown to be constrained to this absolute global minimum, forcing $β= 1/2$ identically and establishing variational evidence supporting a proof of the Riemann Hypothesis (RH). *NOTE: This manuscript Does Not claim to prove the RH. Details on this are covered in the conclusion section.
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