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

This paper constructs a canonical zero-free entire function $\Gamma_{\mathbb{P}}^{\mathrm{cyc}}$ that interpolates the prime Bhargava factorial via cyclotomic and orbitwise normalization, establishing its reflection law, deriving an unsmoothed Stirling formula with a precise prime-local error term, and proving a conditional asymptotic for the sum of squares of this error term.

Original authors: Brian Diaz

Published 2026-07-27

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Brian Diaz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to count the number of ways to arrange a deck of cards. For a standard deck, you multiply $1 \times 2 \times 3 \dots$ all the way up to 52. This is the famous "factorial," a mathematical tool that helps us count combinations, predict probabilities, and understand how things grow. But what if you didn't have a standard deck? What if your deck was made only of prime numbers—those special integers like 2, 3, 5, 7, 11 that can't be broken down further? Mathematicians have long wondered: if we build a "prime factorial" using only these special numbers, does it follow the same smooth, predictable rules as the standard one?

This question lives in the world of number theory, a branch of mathematics that studies the hidden patterns of integers. To answer it, we need two main tools. First, the Gamma function, which is like a "super-factorial" that lets us calculate factorials for numbers that aren't whole numbers (like 3.5!), turning a jagged staircase of integers into a smooth, flowing curve. Second, Stirling's formula, a famous shortcut that tells us how big a factorial gets as the numbers get huge, turning a massive multiplication problem into a simple equation involving logs and square roots. The big mystery has been: when we swap in prime numbers, does this smooth curve still exist, and does the shortcut still work?

In this paper, mathematician Brian Diaz tackles this puzzle by building a brand-new "prime factorial" from the ground up. He doesn't just guess the answer; he constructs a rigorous mathematical machine called a "factorial calculus" that treats prime numbers like layers of a cake. He shows that while each individual layer of prime numbers behaves in a messy, jagged way, when you stack them all together, they form a smooth, predictable whole. He then invents a special "Prime Gamma Function" that acts as the smooth curve for these prime factorials.

The paper's main finding is that this new Prime Gamma Function is real, unique, and follows a beautiful symmetry rule: if you multiply the function at a number $z$ by the function at $1-z$ , you always get 1. This is a "reflection law" similar to the one the standard Gamma function follows, proving that the prime world has its own elegant order. However, the paper also reveals a twist. While the function is perfectly smooth, the difference between the actual prime factorial and the predicted shortcut (called the "Stirling defect") doesn't vanish; it wiggles. The paper proves that these wiggles are real, unsmoothed arithmetic fluctuations, not just errors in calculation.

Diaz calculates exactly how big these wiggles get on average. He finds that if you look at the square of these fluctuations up to a number $X$ , the total grows roughly like $X^{1.5} \log X$ . This is a precise, conditional result. The paper proves that if a specific, difficult hypothesis about the behavior of prime numbers (a "square-root bound" on a complex character sum) is true, then this growth rate is exact. If that hypothesis fails, the growth rate might be different. The paper does not claim to have proven the hypothesis itself; rather, it isolates the entire uncertainty of the problem into this single, well-defined mathematical guess. So, the structure of the prime factorial is proven to be beautiful and unique, but the exact size of its final fluctuations waits on one last piece of the puzzle to fall into place.

Technical Summary: Factorial Calculi and the Canonical Stirling Defect of the Prime Bhargava Factorial

1. Problem Statement and Motivation
The paper addresses the analytic continuation and asymptotic behavior of the prime Bhargava factorial, denoted $(n+1)!_P$ . While the classical Euler gamma function $\Gamma(z)$ is uniquely characterized among solutions to the recurrence $\Gamma(z+1)=z\Gamma(z)$ by the Bohr–Mollerup theorem (positivity and log-convexity), no such direct convexity principle exists for arithmetic factorials defined on subsets of integers.

The specific problem is to construct a "canonical" entire gamma function $\Gamma_P(z)$ that interpolates the prime Bhargava factorial, satisfies its exact recurrence, and possesses a well-defined Stirling expansion. A central difficulty is that the recurrence relation alone admits a "torsor" of entire solutions (differing by 1-periodic entire functions). The paper seeks a normalization principle to select a unique representative, analogous to how Bohr–Mollerup selects the classical gamma function, and to analyze the precise arithmetic fluctuations of this selection.

2. Methodology and Framework

The paper develops a three-part framework:

Part I: The Category of Factorial Calculi.
The authors formalize a category, FactCalc, where objects are positive factorial data sequences $F = (F_n)$ . Each datum defines a weighted Hilbert space of power series and a generalized derivative operator. The prime Bhargava factorial is realized not as a single sequence, but as the filtered symmetric-monoidal completion of "atomic" valuation layers.
- Atomic Layers: For each prime $p$ and integer $j \ge 0$ , a layer $\lambda = (p,j)$ contributes a periodic step factor.
- Completion: The completed factorial $(n+1)!_P$ is the limit of finite products of these layers. The paper proves that while finite layer systems exhibit exponential growth, the completed system exhibits factorial growth, necessitating a shift from disk-based analytic functions to entire functions.
Part II: Gamma-Lift Torsors and Orbitwise Stirling Selection.
The authors analyze the ambiguity in extending the discrete factorial to an entire function.
- Torsor Structure: The set of entire lifts of the logarithmic recurrence forms a torsor under the group of normalized 1-periodic entire functions.
- Normalization Principle: Instead of log-convexity, the paper introduces orbitwise Stirling normalization. This principle requires that the "transverse defect" (the deviation of the lift from a comparison model, averaged over translation orbits) vanishes asymptotically.
- Construction: Using symmetric cyclotomic quadrature, the authors construct a specific entire recurrence symbol $L^{cyc}_P$ and a distinguished lift $H^{cyc}_P$ . This lift is shown to be the unique element of the torsor satisfying the orbitwise Stirling condition relative to the comparison model $M_P(z) = \log \Gamma(z) + C_P(z-1)$ , where $C_P = \sum_p \frac{\log p}{(p-1)^2}$ .
- Properties: The resulting function $\Gamma^{cyc}_P(z) = \exp(H^{cyc}_P(z))$ is zero-free, entire, satisfies the recurrence, and obeys an Euler-type reflection law: $\Gamma^{cyc}_P(z)\Gamma^{cyc}_P(1-z) = 1$ .
Part III: Unsmoothed Fluctuation Analysis.
The paper analyzes the second moment of the "Stirling defect" $R_P(n) = \log(n+1)!_P - \log n! - C_P n$ .
- Decomposition: The defect is decomposed into a geometric diagonal term and off-diagonal covariance terms.
- Reduction: Using Vaughan's identity, Poisson summation, and conductor descent, the authors reduce the variance problem to a "centered subcritical Vaughan moment."
- Conditionality: All structural components (the gamma function, the reflection law, the diagonal constant) are proven unconditionally. The final asymptotic variance depends on a specific conjecture regarding the cancellation of a centered character sum.

3. Key Contributions and Results

Categorical Realization: The prime Bhargava factorial is rigorously defined as the bounded projective limit of atomic valuation calculi, establishing a "factorial calculus" framework that generalizes the classical differential calculus.
Canonical Gamma Function: The construction of $\Gamma^{cyc}_P(z)$ $Γ_{P}^{cy c} (z)$ , the first canonical entire gamma function for the prime Bhargava factorial. It satisfies:
- $\Gamma^{cyc}_P(m) = m!_P$ for integers $m \ge 1$ .
- The reflection law $\Gamma^{cyc}_P(z)\Gamma^{cyc}_P(1-z) = 1$ .
- The midpoint value $\Gamma^{cyc}_P(1/2) = 1$ .
Unsmoothed Stirling Expansion: The paper derives an exact, unsmoothed Stirling formula for the prime factorial:
$\log(n+1)!_P = n \log n + (C_P - 1)n + \frac{1}{2}\log(2\pi n) + R_P(n) + O(1/n)$
where $R_P(n)$ is an explicit arithmetic discrepancy with an absolutely convergent prime-local expansion.
Variance Asymptotic: The paper proves that the second moment of the defect behaves as:
$\sum_{n \le X} R_P(n)^2 \sim -\frac{4}{3}\zeta\left(-\frac{1}{2}\right) X^{3/2} \log X$
This result is conditional on Conjecture 1.6 (Subcritical Centered Vaughan Moment), which asserts a square-root bound for a specific centered character sum involving truncated Möbius and prime weights.
Non-Existence of Positive Integrals: The paper demonstrates that no positive Euler–Mellin representation exists for $\Gamma^{cyc}_P$ or its reciprocal, as the sequence fails Hankel positivity conditions. It motivates a conjecture for a signed, distributional, or contour representation instead.

4. Significance and Claims

The paper claims to provide a complete structural and analytic theory for the prime Bhargava factorial, answering specific questions (Questions 31 and 33) posed by Bhargava in his 2000 work.

Structural Unification: It unifies arithmetic factorials, generalized exponentials, and gamma functions within a single categorical framework of "factorial calculi," distinguishing between formal polynomial conjugacy and analytic boundedness.
Normalization Theory: It establishes orbitwise Stirling rigidity as the correct analog of the Bohr–Mollerup theorem for arithmetic recurrences, selecting a unique analytic continuation without relying on convexity.
Precision of Fluctuations: It isolates the source of uncertainty in the variance of the prime factorial. The paper asserts that the "categorical construction, canonical continuation, reflection law, and Stirling expansion are unconditional," while the variance asymptotic's conditionality is "isolated in a single spectral estimate."
Arithmetic vs. Analytic: The work highlights that the "Stirling behavior" of the prime factorial is a phenomenon of completion; it is not visible in finite layer systems but emerges only after the infinite product of valuation layers is taken.

In summary, the paper constructs a distinguished entire function $\Gamma^{cyc}_P$ that serves as the canonical gamma function for the prime Bhargava factorial, proves its fundamental functional equations, and derives the precise asymptotic variance of its arithmetic error term, contingent only on a specific, narrow conjecture in analytic number theory.

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