Factorial Calculi and the Canonical Stirling Defect of the Prime Bhargava Factorial

Abstract

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 $\displaystyle \Gamma_{\mathbb P}^{\mathrm{cyc}}$ interpolating the prime Bhargava factorial. It satisfies the Euler-type reflection law \[ \Gamma_{\mathbb P}^{\mathrm{cyc}}(z)\Gamma_{\mathbb P}^{\mathrm{cyc}}(1-z)=1, \qquad \Gamma_{\mathbb P}^{\mathrm{cyc}}(1/2)=1. \] At positive integers we obtain the unsmoothed Stirling formula \[ \log (n+1)!_{\mathbb P} =n\log n+(C_{\mathbb P}-1)n+\tfrac12\log(2\pi n) +\mathcal R_{\mathbb P}(n)+O(n^{-1}), \qquad C_{\mathbb P}=\sum_p\frac{\log p}{(p-1)^2}, \] 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 \[ \sum_{n\le X}\mathcal R_{\mathbb P}(n)^2 \sim-\frac43\zeta(-1/2)X^{3/2}\log X. \]