bibkey: osullivan2021modifiedjensen authors: Cormac O’Sullivan year: 2021 title: Zeros of Jensen polynomials and asymptotics for the Riemann xi function doi: 10.1007/s40687-020-00240-5 url: https://arxiv.org/abs/2007.13582v2 claim: Theorem 3.5 and equation (3.6) give an RH criterion using all zero-shift Hermite combinations; the paper’s large-shift sufficient bound does not establish that complete family. strata_touched: [] license: citation-only triage: anchor

The modified Hermite–Jensen criterion

The primary is Cormac O’Sullivan, Zeros of Jensen polynomials and asymptotics for the Riemann xi function, arXiv:2007.13582v2, version submitted 9 December 2020. Journal metadata identifies Research in the Mathematical Sciences 8 (2021), article 46, DOI 10.1007/s40687-020-00240-5. The theorem and equation locators below refer to the arXiv version; the journal metadata does not establish equality of the two texts. This citation-only input and its parameter applications have no new Lean certification or originality claim. No source text is vendored.

Actual coefficients and Hermite convention

Write the source’s positive coefficient $\gamma (k)$ as ${\Gamma }_k$, to distinguish it from the Euler Gamma function. Equations (1.2)–(1.4) define

$$\begin{array}{rl}\Theta (z)& =\xi (1/2+\sqrt{z})=\sum _{k\ge 0}\frac{{\Gamma }_k}{k!}{z}^k,\\ {\Gamma }_k& =\frac{k!}{(2k)!}{\xi }^{(2k)}(1/2),\\ J^{d,n}(X)& =\sum _{j=0}^d\left(\genfrac{}{}{0ex}{}{d}{j}\right){\Gamma }_{n+j}{X}^j,\\ {\mathcal{H}}_{d,n}(X)& =\sum _{j=0}^d\left(\genfrac{}{}{0ex}{}{d}{j}\right){\Gamma }_{n+j}{H}_{d-j}(X).\end{array}\text{\hspace{1em}(MH)}$$

The square-root notation defines an entire function through the even expansion of $\xi$ about $1/2$; it imposes no branch cut. The source calls ${\mathcal{H}}_{d,n}$ by $P^{d,n}$, distinct from the repository’s $P_d$. The physicists’ Hermite convention is

$$e^{2Xt-t^2}=\sum _{m\ge 0}{H}_m(X)\frac{t^m}{m!},H_m(X)=m!\sum _{r=0}^{\lfloor m/2\rfloor }\frac{(-1{)}^r(2X{)}^{m-2r}}{r!(m-2r)!}.\text{\hspace{1em}(HC)}$$

These are O’Sullivan’s coefficients without an additional factor $8$. The actual theta correspondence already supplies ${\Gamma }_k=\Xi (0)k!a_k$, with $a_0=1$, from the complete kernel in Romik’s owning note. That correspondence and the repository’s normalization and primitive interfaces are reused.

The complete zero-shift family already suffices

In section 3.1, genus $1^{\ast }$ means a function of the form $e^{-a{z}^2}f(z)$ with $a\ge 0$ and $f$ entire of genus at most one. Theorem 3.5 assumes a real entire function $F$ of genus $1^{\ast }$ and an auxiliary nonzero Laguerre–Pólya function $\Omega$. It characterizes hyperbolicity of $F$ by hyperbolicity of every reciprocal Jensen polynomial of $F\Omega$. Hyperbolic means that all zeros are real; it does not require simple zeros.

For $F=\Theta$ and $\Omega =e^{-z^2}$, equation (3.6) identifies the degree-$d$ reciprocal polynomial as ${\mathcal{H}}_{d,0}(x/2)$. The source’s genus condition holds for $\Theta$, which is entire of order $1/2$. Theorem 3.5 therefore gives the direct specialization

$$\mathrm{RH}⟺{\mathcal{H}}_{d,0}\text{ is hyperbolic for every integer }d\ge 1.\text{\hspace{1em}(ZF)}$$

Theorem 1.2 states the extended criterion with all $d\ge 1$ and all $n\ge 0$. For the direct route (ZF), a converse implication from an individual modified polynomial to an ordinary Jensen polynomial is unnecessary: the entire zero-shift family is already sufficient.

This family quantifier differs from Corollary 3.8, whose fixed-pair statement is only

$$J^{d,n}\text{ hyperbolic}⟹{\mathcal{H}}_{d,n}\text{ hyperbolic}.\text{\hspace{1em}(FW)}$$

Equations (1.5)–(1.6) exhibit the distinction at degree two:

$$\begin{array}{rl}{J}^{2,n}\text{ hyperbolic}& ⟺{\Gamma }_{n+1}^2\ge {\Gamma }_n{\Gamma }_{n+2},\\ {\mathcal{H}}_{2,n}\text{ hyperbolic}& ⟺{\Gamma }_{n+1}^2+2{\Gamma }_n^2\ge {\Gamma }_n{\Gamma }_{n+2}.\end{array}\text{\hspace{1em}(D2)}$$

The second inequality does not imply the first for general coefficient triples. This algebraic distinction is not a counterexample for the actual xi coefficients and does not contradict the full-family criterion.

Scope of the quantitative sufficient condition

Theorem 1.3 states that, for all sufficiently large $d$, ${\mathcal{H}}_{d,n}$ is hyperbolic whenever

$$\frac{n}{{\log }^{2}n}\ge \frac{d^{3/4}}{2}.\text{\hspace{1em}(LS)}$$

This is a large-shift result, with its expression used for $n>1$. It supplies no application at zero shift and no numerical value for the degree threshold. Moving to a different shift changes the actual coefficient window ${\Gamma }_{n+j}$.

Theorem 8.1, attributed to Turán, supplies real simple roots for $G={\sum }_{j=0}^d{c}_j{H}_j$ under

$$\sum _{j=0}^{d-2}{2}^{j}j!c_j^2<2^d(d-1)!c_d^2.$$

Applied to (MH), equation (8.2) is the sufficient budget

$$\sum _{j=2}^d\frac{d}{2^{j}j!}\left(\genfrac{}{}{0ex}{}{d}{j}\right){\left(\frac{{\Gamma }_{n+j}}{{\Gamma }_n}\right)}^2<1.\text{\hspace{1em}(TB)}$$

For fixed actual $n\ge 0$ and $d\ge 2$, its positive $j=2$ term is

$$\frac{d^2(d-1)}{16}{\left(\frac{{\Gamma }_{n+2}}{{\Gamma }_n}\right)}^2.\text{\hspace{1em}(FT)}$$

The ratio is fixed and positive, so (FT) grows without bound as $d$ increases. Thus (TB) cannot prove unbounded-degree hyperbolicity at a fixed shift, including zero. This is a limitation of this sufficient condition, not nonhyperbolicity of the family or a disproof of RH.

Exact finite-polynomial correspondence

Reuse the normalized Jensen tower with $P_d(X)=J^{d,0}(X/d)/\Xi (0)$ and its monic reflection $q_d(x)=x^d{P}_d(-1/x)$. For $D_x=d/dx$, (HC) gives the finite-polynomial parameter application

$$\frac{{\mathcal{H}}_{d,0}(-dx/2)}{\Xi (0)(-d{)}^d}=\exp (-D_x^2/d^2)q_d(x),d\ge 1.\text{\hspace{1em}(HT)}$$

Here the exponential is the finite sum ${\sum }_{r=0}^{\lfloor d/2\rfloor }(-1{)}^r{D}_x^{2r}/(r!d^{2r})$ on degree-$d$ polynomials. Its sign is part of the convention; its inverse need not preserve real roots. Equation (HT) identifies a polynomial transform, not a deformation of the complete Xi kernel at a physical de Bruijn–Newman time.

For a monic $q_d$ and $d\ge 2$, the $x^{d-1}$ coefficient stays fixed and the $x^{d-2}$ coefficient changes by $-(d-1)/d$. The parameter $1/d^2\to 0$ consequently gives no vanishing coefficient correction uniformly across these increasing degrees, and by itself gives no uniform control of their roots.

Equation (ZF) leaves a concrete alternative obligation: control the actual modified zero-shift family for every degree. A fixed-degree inference back to $q_d$ would require a separate reverse estimate; the direct full-family route does not. The source statements and these parameter applications supply neither that all-degree estimate nor the complete signed Robin-tail bound.

Source display limitation

The inspected HTML’s intermediate expansion of $g_d^{\ast }(e^{-z^2};x)$ after Theorem 3.5 displays the power $x^{d-r}$. The definition of the reciprocal Jensen polynomial and (HC) require $x^{d-2r}$. Equation (3.6) supplies the intended Hermite identity; the inconsistent intermediate powers are not used. This observation concerns the HTML display and does not identify an error in the journal text or establish a PDF comparison.

Scope of the finite Hermite and Appell suppliers

Antonio J. Durán’s Zeros of Linear Combinations of Orthogonal Polynomials, Mediterranean Journal of Mathematics 23, article 148, published 18 June 2026, DOI 10.1007/s00009-026-03145-9, gives a finite-band supplier in Corollary 4.1 and section 4.1, Remark 1. For fixed $K\ge 2$, real coefficients $c_0=1$, $c_K\ne 0$, and $C_K={\max }_{2\le j\le K}|c_j|$, the combination

$$U_m(X)=\sum _{j=0}^K{c}_j{H}_{m-j}(X)$$

has real simple zeros interlacing those of $H_{m-1}$ under

$$m\ge \max \{2K,(2^{K-1}-1{)}^2{4}^{K-1}{C}_K^2\}.\text{\hspace{1em}(DB)}$$

The separate Hermite preprint, Zeros of linear combinations of Hermite polynomials, arXiv:2505.15330v1, submitted 21 May 2025, states the smaller coefficient factor $(K-1{)}^2{4}^{K-2}$ in Corollary 1.1. These explicit bounds have different factors and both require $m\ge 2K$; they are not interchangeable versions of the same numerical estimate.

Matching the full actual unnormalized family requires

$$m=K=d,c_j=\left(\genfrac{}{}{0ex}{}{d}{j}\right)\frac{{\Gamma }_j}{{\Gamma }_0}.\text{\hspace{1em}(DM)}$$

Since ${\Gamma }_d>0$, the last Hermite coefficient cannot be dropped. The condition $m\ge 2K$ therefore excludes every target diagonal $d\ge 2$. This excludes these sufficient bounds, not real-rootedness of the actual family.

The Hermite preprint’s Theorem 1.2 also has a stronger branch: if $P_K(X)={\sum }_{j=0}^K{c}_j{X}^{K-j}$ already has only real zeros, the polynomials $U_m$ have real simple zeros for $m\ge K$, and the zeros of $U_{m+1}$ interlace those of $U_m$. At (DM), the coefficient polynomial is $X^d{J}^{d,0}(1/X)/{\Gamma }_0$. Thus that branch consumes ordinary Jensen real-rootedness; it does not bypass the actual zero-shift sign obligation. Its other branch gives an eventual threshold depending on one fixed coefficient polynomial and supplies no uniform threshold along the changing actual diagonal.

Theorem 1.3 of the same preprint instead uses $h_m=H_m/(2^{m}m!)$ and one fixed finite coefficient polynomial. Exact normalization of the actual family gives

$$\frac{{\mathcal{H}}_{d,0}(X)}{{\Gamma }_0{2}^{d}d!}=\sum _{j=0}^d\frac{a_j}{2^j}{h}_{d-j}(X),R(z)=\sum _{j\ge 0}\frac{a_j}{2^j}{z}^j=\frac{\Theta (z/2)}{{\Gamma }_0}.\text{\hspace{1em}(DA)}$$

Here the coefficients are fixed but the actual generating factor $R$ is infinite. A fixed finite truncation changes the source; taking $K=d$ restores each individual polynomial without supplying a degree-uniform threshold. Local uniform convergence of truncations alone does not bound the omitted coefficients against all moving root or critical-value margins. No such actual-source tail estimate is supplied by this finite-polynomial theorem.

Durán’s Brenke polynomials with real zeros and the Riemann Hypothesis, arXiv:2405.18940v1, supplies the established Appell–Laguerre–Pólya equivalence in Theorem A. Its Theorem 1.4 explicitly assumes $B\in \mathrm{LP}\mathrm{I}$ in addition to a coefficient-ratio limit. When $B$ is the normalized actual theta source, that LP hypothesis retains an RH-linked obligation; the ratio limit cannot replace it.

These are literature applications with pinned hypotheses. They supply no all-degree actual zero-shift estimate, signed Robin-tail budget, new mathematical declaration or Lean certification.

Known finite-degree coverage from verified zero height

Let ${\mathrm{RH}}_0(T)$ mean that every zero $\rho$ of $\xi$ with $|\mathrm{Im}\rho |\le T$ lies on the critical line. This requires complete zero counting, not only a list of zeros found on that line. Griffin, Ono, Rolen, Thorner, Tripp and Wagner, Jensen polynomials for the Riemann xi-function, arXiv:1910.01227v3, submitted 17 December 2020, use exactly $\gamma (j)={\Gamma }_j$ in equations (1.1)–(1.2). Their Theorem 1.2 at $m=0$ gives

$${\mathrm{RH}}_0(T)⟹J^{d,n}\text{ is hyperbolic for every }n\ge 0\text{ and }1\le d\le \lfloor T{\rfloor }^2.\text{\hspace{1em}(VH)}$$

The source identifies this specialization as Chasse’s Theorem 1.8. Its Lemmas 5.1–5.2 supply differentiation closure of the canonical sector class and the finite-degree sector implication; no new zero-pair preserver proof is needed for (VH).

Platt–Trudgian’s The Riemann hypothesis is true up to $3\cdot 10^{12}$, arXiv:2004.09765v1, submitted 21 April 2020, Theorem 1 supplies a rigorous complete interval-arithmetic verification through a height exceeding the conservative bound $T=3\cdot 10^{12}$. Applying (VH) and the existing implication (FW) therefore covers both actual families:

$$J^{d,n}\text{ and }{\mathcal{H}}_{d,n}\text{ are hyperbolic}(n\ge 0,1\le d\le 9\cdot 10^{24}).\text{\hspace{1em}(FC)}$$

O’Sullivan explicitly reports this extension in introductory footnote 1. Griffin et al.’s printed Corollary 1.3 instead has the older bound $9.36\cdot 10^{20}$; (FC) uses their theorem with the later verification. This is direct reuse of established coverage, not a new estimate or a replay of the numerical certificate. For the zero-shift criterion (ZF), degrees above $9\cdot 10^{24}$ remain outside this finite input. It supplies no all-degree conclusion, signed Robin-tail budget or local Lean certification.