A Local Toeplitz--Hankel Criterion Equivalent to the Riemann Hypothesis

Abstract

Let $\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the completed Riemann xi-function, and let $F(x)=\frac{\xi'}{\xi}\!\left(\frac{1}{1-x}\right)=\sum_{n\ge 0} f_nx^n$ initially denote its germ at the origin. We introduce the real symmetric Toeplitz--Hankel matrix $c_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0, \qquad i,j\ge 0.$ We prove that the Riemann hypothesis is equivalent to positive semidefiniteness of every finite leading principal block of this matrix. More strongly, the full matrix condition is equivalent, for the purpose of testing the Riemann hypothesis, to only the adjacent local inequalities $2f_0-f_{2n+1}\ge 0,$ $(2f_0-f_{2n+1})(2f_0-f_{2n+3})\ge (f_1-f_{2n+2})^2 \qquad(n\ge 0).$ The converse implication uses a Pringsheim bootstrap: these local inequalities force the germ of $F$ to have Taylor radius at least one, hence exclude zeros of $\xi$ from the half-plane $\Re s>1/2$. If $\lambda_n$ are the Keiper--Li coefficients, then $f_n=\lambda_{n+1}-2\lambda_n+\lambda_{n-1}$, so the criterion is a local quadratic condition on their second finite differences. This is an equivalent reformulation, not a proof of the Riemann hypothesis. To the best of our knowledge, the exact Toeplitz--Hankel matrix and its reduction to adjacent $2\times2$ conditions have not appeared previously.