Bhargava Gamma functions for determinant-admissible sets

Technical Summary: Bhargava Gamma Functions for Determinant-Admissible Sets

Problem Statement
M. Bhargava's theory of generalized factorials assigns a sequence of factorial ideals to an arbitrary subset $S$ of a Dedekind domain $R$. While these ideals unify concepts like fixed divisors and integer-valued polynomials, the existence of a corresponding "Gamma function" and Stirling's formula for general $S$ remains an open problem. Previous work, such as Lamoureux's analysis of the norm of the Pólya–Ostrowski factorial ideal, provided asymptotic formulas but did not identify the underlying local dynamical systems or a unified completion framework. The central problem addressed is identifying the appropriate analytic object that completes the discrete factorial recurrence $[n]!_S$ for general subsets, particularly in cases where the factorial is not a principal ideal and thus lacks a canonical scalar generator.

Methodology
The paper proposes a "determinant-completion theory" where the Gamma object is not an arbitrary analytic interpolation of values, but a completion of local ordering dynamics. The methodology proceeds through the following structural chain:

1. Local Ordering Dynamics: For each prime $p$, the $p$-ordering of $S$ generates a recurrence cocycle $a_{S,p}(n)$. The paper models these dynamics using state spaces, transfer operators, and renewal systems (carry, orbit, or renewal models).
2. Relative Determinants: Instead of treating the recurrence as a scalar sequence, the author defines a "local determinant admissibility" condition. This requires the existence of a normalized relative determinant section $\Delta_{S,p}(z)$ derived from the intrinsic transfer data of the local system.
3. Global Assembly: A global determinant datum is constructed by combining locally admissible systems at finite places with a primitive determinant line at infinite places (Archimedean or function-field infinity).
4. Line-Valued Interpolation: The resulting object, the "Bhargava Gamma line," is a meromorphic section of a line bundle. A scalar Gamma function is only obtained after applying a norm, choosing a principalization, or taking a root of the line. This framework naturally handles cases where factorial ideals are non-principal.

Key Contributions and Results

• Determinant-Admissibility Framework: The paper defines precise local and global hypotheses (Definitions 3.3 and 4.1) under which a subset $S$ admits a Gamma completion. It establishes that the Gamma object is generally a section of a determinant line rather than a scalar function.
• Stirling and Reflection as Spectral/Dual Phenomena:
  • Stirling's Formula: The paper reframes Stirling's asymptotic expansion as the large-parameter expansion of the completed determinant metric. The finite-place remainder is identified as a "centered renewal determinant" rather than an unexplained error term.
  • Reflection Formulas: Euler reflection is shown to arise from duality isomorphisms between local and infinite-place determinant complexes, rather than being an automatic property.
• Classification of Examples: The theory recovers and unifies several known cases:
  • Ordinary Factorial: Recovered as a scalar function with a single Archimedean spectral line.
  • Quadratic Images: Recovers Takeda's exact formulas as shifted Gamma factors multiplied by finite cyclotomic determinants.
  • Geometric Progressions: Provides a Fredholm determinant completion for $q$-factorials, linking them to Jackson's Gamma function.
  • Global Rings: For the full ring of integers $O_K$, the theory recovers Lamoureux's Stirling coefficients and interprets the Pólya group obstruction as a non-trivial line class.
  • Function Fields: Identifies the Carlitz factorial as a scalar trivialization of a place-completed determinant line, explaining log-periodic fluctuations via the poles of the zeta function.
• The Cubic-Power Set ($C = \{m^3\}$): This is developed as the primary non-trivial example.
  • For primes $p \equiv 1 \pmod 3$, the local dynamics admit an exact three-state carry determinant.
  • Character projection yields an Archimedean "half-determinant" $\Gamma_{C,\infty}(z) = \sqrt{\Gamma(z)\Gamma(3z-2)}$, which is a section of a square-root line.
  • The paper isolates the "inert-prime renewal system" ($p \equiv 2 \pmod 3$) and the ramified system at $p=3$ as the source of the finite correction.
  • A candidate normalization constant $\beta_C$ is proposed, separating ramified, Galois, and nonlinear inert-prime contributions.

Status of the Work and Open Problems
The paper is presented as a working draft (v0.5) that rigorously develops the algebraic formalism and exact finite-state examples (specifically the cubic-residue sector). However, it explicitly acknowledges that the "principal analytic problem" remains unsolved:

• The canonical analytic continuation of the centered determinant for the inert-prime renewal system ($p \equiv 2 \pmod 3$) has not been constructed.
• A global uniqueness theorem excluding integer-vanishing entire factors for the cubic case is not yet proven.
• The multivariable theory is explicitly excluded, noted as a separate problem involving different indexing geometry (Vandermonde determinants of multivariate matrices).

Significance
The paper argues that the correct generalization of the Gamma function for Bhargava factorials is not a single universal scalar function, but a "place-completed determinant object" attached to local ordering dynamics. The scalar function, when it exists, is merely a "shadow" of this line-valued object. This perspective shifts the focus from interpolation to the spectral and dynamical properties of the underlying $p$-orderings. The framework successfully explains why some sets admit scalar Gamma functions while others (like the cubic set or non-Pólya fields) require line-valued or root-line objects, and it provides a structural explanation for the appearance of Stirling coefficients and reflection formulas in terms of spectral asymptotics and duality.