AI 中文总结
本文通过阶乘算术等方法，构建了一个规范零自由整个函数Γ_P^cyc，满足欧拉型反射定律，并给出了素数阶乘的斯特林公式及Hankel正性相关结论。
AI 中文摘要
经典伽马函数通过欧拉递推关系被Bohr-Mollerup定理所突出。我们为附属于有理素数的Bhargava阶乘发展了类似的规范理论。利用阶乘算术、过滤素层完成、中心环状四次方程和轨道wise斯特林规范，我们构造了一个可以插值素数Bhargava阶乘的规范零自由整个函数Γ_P^cyc。它满足欧拉型反射定律Γ_P^cyc(z)Γ_P^cyc(1-z)=1，Γ_P^cyc(1/2)=1。在正整数处，我们得到未平滑的斯特林公式log(n+1)!_P =n log n + (C_P -1)n + 1/2 log(2πn) + R_P(n) + O(n^{-1})，其中C_P = Σ_p log p/(p-1)^2，R_P(n)=o(n)且具有精确的素数局部分数部分展开。这给出了Bhargava论文《阶乘函数及其推广》中问题31和33的素数情况回应。Hankel正性排除了函数及其倒数的正欧拉-梅林表示；环状构造反而促使了一个素数Hankel猜想，关于该函数倒数的规范轮廓或分布性表示。最后，除了一个中心亚临界Vaughan特征矩外，所有情况均无条件控制。假设相应的平方根估计，我们证明Σ_{n≤X} R_P(n)^2 ~ -4/3 ζ(-1/2) X^{3/2} log X。
英文摘要
The classical gamma function is singled out among solutions of Euler's recurrence by the Bohr--Mollerup theorem. We develop an analogous normalization theory for the Bhargava factorial attached to the rational primes. Using factorial calculi, filtered prime-layer completion, centered cyclotomic quadrature, and orbitwise Stirling normalization, we construct a canonical zero-free entire function $\displaystyle Γ_{\mathbb P}^{\mathrm{cyc}}$ interpolating the prime Bhargava factorial. It satisfies the Euler-type reflection law \[ Γ_{\mathbb P}^{\mathrm{cyc}}(z)Γ_{\mathbb P}^{\mathrm{cyc}}(1-z)=1, \qquad Γ_{\mathbb P}^{\mathrm{cyc}}(1/2)=1. \] At positive integers we obtain the unsmoothed Stirling formula \[ \log (n+1)!_{\mathbb P} =n\log n+(C_{\mathbb P}-1)n+\tfrac12\log(2πn) +\mathcal R_{\mathbb P}(n)+O(n^{-1}), \qquad C_{\mathbb P}=\sum_p\frac{\log p}{(p-1)^2}, \] where $\displaystyle \mathcal R_{\mathbb P}(n)=o(n)$ and admits an exact prime-local fractional-part expansion. This gives a prime-case response to Questions 31 and 33 of Bhargava's paper \emph{The Factorial Function and Generalizations}. Hankel positivity rules out positive Euler--Mellin representations for the function and its reciprocal; the cyclotomic construction instead motivates a prime Hankel conjecture for a canonical contour or distributional representation of the reciprocal. Finally, all but one centered subcritical Vaughan character moment are controlled unconditionally. Assuming the corresponding square-root estimate, we prove \[ \sum_{n\le X}\mathcal R_{\mathbb P}(n)^2 \sim-\frac43ζ(-1/2)X^{3/2}\log X. \]
