bibkey: liu2026xilogconcavity
authors: Yanxin Liu; Jianxi Mao
year: 2026
title: Infinite log-concavity of the Taylor coefficients of the Riemann xi-function
doi: null
url: https://arxiv.org/abs/2610.04972v1
claim: The preprint states unconditional strict infinite log-concavity of the central Taylor coefficients of the Riemann xi-function; it does not supply the Laguerre--Pólya conclusion or a pointwise Robin estimate.
strata_touched: []
license: citation-only
triage: anchor

Infinite log-concavity of the Taylor coefficients of the Riemann xi-function

The source is arXiv:2610.04972v1, submitted 4 October 2026. The result is a preprint and was not independently audited or formalized here.

Exact result and logical direction

Write

$$F(x)=\xi \left(\frac{1}{2}+\sqrt{x}\right)=\sum _{n\ge 0}{\lambda }_n{x}^n,$$

and let $L$ be the standard log-concavity operator, $(La{)}_n=a_n^2-a_{n-1}{a}_{n+1}$, with $a_{-1}=0$. Theorem 1.1 states

$$(L^m\lambda {)}_n>0(m,n\ge 0).$$

The paper combines saddle estimates, a two-index recurrence for logarithmic ratios, interval verification for the finite initial range, and a global closure argument. The introduction records the classical implication that RH would put $F$ in the Laguerre–Pólya class and hence imply infinite log-concavity; the new theorem proves the coefficient property without assuming RH.

Correspondence with the fixed theta measure

To distinguish these coefficients from the prescribed Jensen node parameter, write $b_j={\lambda }_j={\xi }^{(2j)}(1/2)/(2j)!$. With the Romik source normalization, put

$$T=x^2/4,S(T)=\Phi (2\sqrt{T}),d\mu (T)=\frac{2S(T)}{{\xi }_0\sqrt{T}}dT,{\xi }_0=\xi (1/2).$$

Equations (4.3)–(4.4), printed p.43, give the positive-kernel moment representation of $b_j$. The paper’s kernel in (4.3) is $2\Phi (x)$; it is not the Csordas-normalized $K(u)$ used in the square-variable concavity note. Writing $M_j={\int }_0^{\infty }{T}^{j}d\mu (T)$ gives the exact correspondence

$$b_j=\frac{{\xi }_0{4}^j}{(2j)!}{M}_j.$$

This is reuse of the source representation in the fixed receiving coordinate, not an additional coefficient theorem. Theorem 1.1 concerns the ordinary Taylor coefficients $b_j$. The Jensen convention uses ${\gamma }_j=j!b_j$ in the paper’s equation (1.1); its factorial multiplier cannot be discarded when applying an iterated log-concavity conclusion. The coefficient result does not supply the signed neighboring-row comparison at every prescribed node or the extra vanishing at common zeros. Those interfaces remain separate obligations.

Boundary for Robin and FIB ATOM

The coefficient sequence is a central Taylor expansion of $\xi$, whereas Robin’s target is a pointwise divisor-sum inequality. These ${\lambda }_n$ are not the Li coefficients. The stated result supplies no implication to positivity of all Li coefficients, a zero-free half-plane, or a bound on the same integer’s signed Robin tail. Neither this source nor the five-window FIB definitions supply a coefficient-preserving map from $N_g=1+F_{r}g$ to the Taylor index and coefficient ${\lambda }_n$.

Reuse the stated coefficient property when its hypotheses and conclusion match the task; it is not a new Robin estimate. Any application to the Robin/FIB route still needs a proved map preserving the actual target and its quantifiers. The preprint does not supply that map.