bibkey: bober2009factorialratios
authors: Jonathan W. Bober
year: 2009
title: Factorial ratios, hypergeometric series, and a family of step functions
doi: 10.1112/jlms/jdn078
url: https://arxiv.org/abs/0709.1977v1
claim: Theorem 1.2 classifies balanced integral factorial ratios of height one; Section 1.1 relates the associated step functions to the existing Nyman-Beurling criterion, without an unconditional Robin tail estimate.
strata_touched: []
license: citation-only
triage: anchor

Factorial-ratio step functions and the existing RH criterion

The primary is arXiv:0709.1977v1, with arXiv version stamp 13 September 2007. The 23-page PDF has SHA-256 5f3cb3b1e7f74b791608669ce22a6d07ab51faac68436d806c9b329c9abf3201. The publication is Journal of the London Mathematical Society 79 (2009), 422–444, DOI 10.1112/jlms/jdn078. The source locators below identify inspected statements and their scope; the complete classification proof has not been independently audited or Lean compiled. No source text or proof is vendored.

Integrality and the classified family

For positive integer tuples $a=(a_1,\dots ,a_K)$ and $b=(b_1,\dots ,b_L)$, write

$$u_n(a,b)=\frac{{\prod }_{i=1}^K(a_{i}n)!}{{\prod }_{j=1}^L(b_{j}n)!},{\Phi }_{a,b}(u)=\sum _i\lfloor a_{i}u\rfloor -\sum _j\lfloor b_{j}u\rfloor .$$

Landau’s criterion, stated in Theorem 3.1 and specialized in Lemma 3.2, PDF p.7, equates integrality of every $u_n$ with nonnegativity of ${\Phi }_{a,b}$ on $[0,1]$. Under the balance condition ${\sum }_i{a}_i={\sum }_j{b}_j$, this step function is periodic. Lemma 3.4, PDF p.8, relates its reflected values and its height $L-K$. These are literature suppliers, not new project results.

Theorem 1.2, PDF p.3, classifies the case $L=K+1$, after canceling common numerator and denominator parameters and imposing $\gcd (a_1,\dots ,a_K,b_1,\dots ,b_L)=1$. There are three displayed families, (7)–(9), and 52 sporadic parameter sets in Table 2. This is a height-one classification, not a classification of all nonnegative factorial-ratio kernels or their signed linear combinations.

The paragraph surrounding (10), PDF p.3, records that the best Chebyshev constants obtained by the indicated argument from this family are approximately $0.92$ and $1.11$. Its example is the classical ratio

$$\frac{(30n)!n!}{(15n)!(10n)!(6n)!}.$$

That ratio already has the repository divisibility developments BalaChebyshevThreeDivisibility and BalaChebyshevFiveDivisibility. Their presence is a reuse locator; no current build or classification formalization is asserted here.

The RH connection is already Nyman-Beurling approximation

Theorem 1.1 and Lemma 3.5 relate the height-one ratios to step functions

$$f(x)=\sum _k{c}_k\lfloor x/m_k\rfloor ,m_k\in {\mathbb{N}}_{>0},c_k\in \mathbb{Z},\sum _k\frac{c_k}{m_k}=0,$$

with values in $\{0,1\}$. Section 1.1, PDF pp.4–5, places these functions in the existing Nyman-Beurling problem. After $x=1/u$, the finite balanced combinations correspond to fractional-part probes. The criterion requires approximation of the constant function in $L^2((1,\infty ),dx/x^2)$; equivalently, one can use the corresponding unit-interval formulation. The complete natural-dilation criterion is already sourced in Baez-Duarte’s strengthening. A factorial-ratio integrality statement does not supply that norm convergence. Bober’s Theorems 1.3 and 1.5, PDF p.4, use a linear span with arbitrary real coefficients (or its equivalent complexification), unlike the integer coefficients in the classified blocks above. Coefficients can change with the approximation stage. This criterion does not assert that height-one functions alone suffice. Stagewise combinations of bounded blocks need not themselves be uniformly bounded.

The specific natural-prefix nonconvergence result and its limitations concerning subsequences are already recorded in Baez-Duarte 2000. Bober’s separate divergence citation on PDF p.5 is reference [2], A divergent Vasyunin correction, identified on p.22 as arXiv:math/0506318v1; it is not Proposition 4.4 of the 2000 paper. The latter obstruction must not be widened to exclude every factorial-ratio construction, every subsequence, or freely chosen coefficients. Pointwise convergence alone does not supply the needed norm estimate without suitable global control; a uniformly bounded sequence on this finite-measure space is a different situation, where dominated convergence does apply.

Remaining obligation for the ordinary Robin source

The joint ordinary source fixes the same actual integer $N$ and $A=\log N$, with $\psi (t)={\sum }_{p^k\le t}\log p$. The target remains an eventual uniform lower bound for

$$\sqrt{A}\log A{\int }_A^{\infty }(\psi (t)-t)\frac{1+\log t}{t^2{\log }^{2}t}dt.$$

All ordinary prime powers, the actual exponent vector and the complete CA reserve must be retained together. Neither the height-one classification, the quoted Chebyshev constants, nor the criterion supplies this signed estimate on that source. A factorial-kernel route would still need a quantitative transfer with its whole residual paid at this scale. The existing comparison analysis and actual FIB target support analysis remain the relevant interfaces; a positive branch classification is not an estimate for their signed residual. No general impossibility theorem for factorial methods or new analytic bound is asserted.