arXiv subjects

Brian Diaz

Publications and source records attributed to Brian Diaz.

4 recordsLinked to original sources

arXiv2026

Bhargava Gamma functions for determinant-admissible sets

Bhargava's factorial construction attaches to a subset $S$ of a Dedekind domain a sequence of factorial ideals determined by local $p$-orderings. We introduce determinant-admissibility, a framework for factorial calculi whose local ordering data admit canonical carry, orbit, or renewal models together with compatible relative determinants. For such sets, the natural analogue of the Gamma function is generally not a scalar-valued function but a completed determinant line. After clearing rational spectral multiplicities, its integer fibers recover a finite tensor power of the Bhargava factorial ideals. Over number fields this line carries Archimedean metrics, while over global function fields it is completed at the distinguished places at infinity. In this formulation, Stirling-type formulas arise from large-parameter asymptotics of metrized determinants, and reflection formulas arise from dualities of the corresponding determinant complexes. We develop the theory through classical factorials, quadratic polynomial images, geometric and $q$-factorials, Polya--Ostrowski factorial ideals, Carlitz--Goss factorials, and a Bhargava--Barnes lift whose second-level determinant recovers the optimal Vandermonde energy of $n$-optimal sets. The set of cubes provides the first genuinely renewal-theoretic example. For primes $p \equiv 1 \pmod{3}$, we construct an exact three-state carry determinant, while projection through the quadratic character $\chi_{-3}$ yields the Archimedean half-determinant \[ \Gamma_{C,\infty}(z) = \sqrt{\Gamma(z)\Gamma(3z-2)} \] and a natural square-root determinant line. The remaining finite-place contribution is governed by nonlinear renewal systems at primes $p \equiv 2 \pmod{3}$ and by a ramified system at $p=3$; constructing their canonical global relative determinant is the principal open analytic problem.

math.NT↗

arXiv2026

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

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 \[ \sum_{n\le X}\mathcal R_{\mathbb P}(n)^2 \sim-\frac43\zeta(-1/2)X^{3/2}\log X. \]

math.NT↗

arXiv2026

Dirichlet Series and Asymptotics for Generalized Legendre Factorials

Let $K$ be a fixed number field, let $\Sigma$ be a finite set of nonzero prime ideals of $\mathcal{O}_K$, and let $f$ be a positive integer-valued function on the prime ideals outside $\Sigma$. We study the ideal-valued factorial defined by $v_{\mathfrak p}(n!_{K,f,\Sigma})=\sum_{k\geq 0}\left\lfloor \frac{n}{f(\mathfrak p)(\mathrm{N}\mathfrak p)^k}\right\rfloor$. Assume that $f(\mathfrak p)=c\,\mathrm{N}\mathfrak p+O((\mathrm{N}\mathfrak p)^{1-\delta})$ for some $c>0$ and $\delta>0$. We derive a Dirichlet series for the logarithmic increments and compare its local prime-power sequence with the ordinary prime-ideal von Mangoldt sequence. A prime-ideal theorem and a Dirichlet hyperbola argument then give $$\log \mathrm{N}_{\Sigma}(n!_{K,f,\Sigma})=c^{-1}n\log n+C_{K,f,\Sigma}n+O(ne^{-a\sqrt{\log n}})$$ for some $a>0$. The linear coefficient is explicit: $C_{K,f,\Sigma}=c^{-1}(\gamma+\kappa_{K,\Sigma}-\log c-1)+J_{K,f,\Sigma}(1)$, where $\kappa_{K,\Sigma}$ is the constant term in the Laurent expansion of $-\zeta'_{K,\Sigma}(s)/\zeta_{K,\Sigma}(s)$ at $s=1$, and $J_{K,f,\Sigma}$ is an absolutely convergent prime-ideal correction near $s=1$. The method also applies to factorial ideals of Legendre subsets whose local class numbers have the corresponding geometric form.

math.NT↗

arXiv2019

Extending a catalog of mock and quantum modular forms to an infinite class

Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed a catalog of mock modular forms $V_{mn}$ having weight $3/2$ eta-theta function shadows and showed that these mock modular forms when viewed on certain sets of rationals, transform as quantum modular forms under the action of explicit subgroups. In this paper, we introduce an infinite class of functions that generalizes one row of the catalog, namely the $V_{m1}$, and show that the functions in this infinite class are both mock modular and quantum modular forms.

math.NT↗