Start with the positive real-axis weight

$$ d\mu(x)=Z^{-1}\frac{dx}{\xi(1/2+x)} =Z^{-1}e^{-Q(x)}\,dx, \qquad Q(x)=\log \xi(1/2+x). $$

It has moments of every order, so it produces orthonormal polynomials \(p_n\), Jacobi coefficients \(b_n>0\), Hankel determinants, and an \(n\)-particle orthogonal-polynomial ensemble. Evenness deletes the diagonal recurrence term:

$$ xp_n(x)=b_{n+1}p_{n+1}(x)+b_np_{n-1}(x). $$

That single recurrence is the skeleton. The lovely thing about this paper is how much anatomy gets hung on it: finite-index lower bounds, a sharp shifted asymptotic, eventual monotonicity, Gaussian fluctuations, an exact variance identity, a global particle law, and an \(n^2\) partition-function law. It is not one asymptotic wearing several hats. The conclusions come from different pieces of structure, and they fit together almost suspiciously well.

Before any limit, the ensemble itself is exact. Its joint density is proportional to

$$ \prod_{1\leq j

The Vandermonde square supplies the particle repulsion; reciprocal \(\xi\) supplies the confinement. If

$$ F(t)=\int_{\mathbb R}\frac{e^{itx}}{\xi(1/2+x)}\,dx, $$

then Andreief's identity identifies the ensemble partition function with the confluent determinant

$$ H_n(t)=(-1)^{n(n-1)/2} \det\!\left[F^{(j+k)}(t)\right]_{j,k=0}^{n-1}. $$

At \(t=0\), the same object factors through the orthogonal-polynomial norms and hence through the recurrence coefficients. This is the paper's useful closed circuit: particles, Jacobi matrices, Fourier derivatives, and Hankel determinants are not analogies for one another. They are exact presentations of one object.

A floor at every index#

Asymptotics usually tell us what happens after politely refusing to say when. Here there is also a bound for every \(n\).

Let \(a_n>0\) solve the virial equation

$$ a_nQ'(a_n)=2n-1. $$

Then

$$ b_n\geq \frac{n}{Q'(a_n)} =\frac{na_n}{2n-1}, $$

and more explicitly,

$$ b_n>\frac{2n}{\log(4n-2)} \qquad(n\geq2). $$

There is even a global floor:

$$ b_n^2\geq b_1^2 \qquad(n\geq1). $$

The mechanism is unusually tidy. For \(Q=\log\xi(1/2+x)\), the classical Riemann density supplies

$$ Q''(x)>0,\qquad Q'''(x)\leq0\quad(x>0). $$

Those signs make \(Q'(x)^2\) a concave function of the virial variable \(xQ'(x)\). Integration by parts gives both the coefficient identity involving \(n/b_n\) and the exact expectation

$$ \int xQ'(x)p_{n-1}(x)^2\,d\mu(x)=2n-1. $$

Cauchy-Schwarz and Jensen then land directly on the lower bound. I like this proof because \(2n-1\) is not decorative bookkeeping: it is the precise hinge between the polynomial degree and the potential.

The \(7/8\) survives#

The leading recurrence scale is already a cute Lambert-\(W\) object:

$$ b_n\sim \frac{\pi n}{W_0(2n/e)}. $$

But the paper retains the first shift:

$$ b_n= \frac{\pi(n-7/8)} {W_0\!\left(2(n-7/8)/e\right)} +O\!\left(\frac{\log n}{n}\right). $$

The fraction \(7/8\) comes from the completed-zeta expansion

$$ Q'(x)=\frac12\log\frac{x}{2\pi} +\frac{7}{4x} +\frac{1}{48x^2} +O(x^{-3}). $$

Inserted into the Mhaskar-Rakhmanov-Saff equation, the \(7/(4x)\) term integrates to \(7/8\). That shift would still be lost if the recurrence coefficient were known only to relative error \(O(1/n)\). The fussy part is a cancellation in the \(\rho=0\) Riemann-Hilbert recurrence functional: the complete first-order endpoint contribution vanishes, improving the comparison with the MRS radius \(R_n\) to

$$ b_n=\frac{R_n}{2}\left(1+O(n^{-2})\right). $$

So the tiny constant makes it all the way from the gamma factor, through the equilibrium scale, through endpoint parametrices, and into \(b_n\). That is the detail I keep staring at.

The shifted law also gives

$$ \frac{n}{W_n}\left(\frac{\pi n}{b_n}-W_n\right)\longrightarrow\frac78, \qquad W_n=W_0(2n/e), $$

along with

$$ n\left(\frac{b_{n+1}}{b_n}-1\right)\longrightarrow1, \qquad (1+W_n)(b_{n+1}-b_n)\longrightarrow\pi. $$

In particular, the recurrence coefficients are eventually strictly increasing. The all-index floor and eventual monotonicity are different statements: one prevents a collapse anywhere, while the other describes the tail's actual direction.

The ensemble fluctuates#

Let \(X_1,\ldots,X_n\) be the particles, and for a real polynomial \(P\) set

$$ L_n(P)=\sum_{j=1}^nP(X_j/b_n). $$

If

$$ \widehat P_k=\frac1{2\pi}\int_0^{2\pi} P(2\cos\theta)e^{-ik\theta}\,d\theta, $$

then every centered polynomial statistic has the limit

$$ L_n(P)-\mathbb{E}L_n(P) \Longrightarrow N\!\left(0,\sum_{k\geq1}k\widehat P_k\widehat P_{-k}\right). $$

The reason is local in index: for every fixed \(r\), \(b_{n+r}/b_n\to1\), so the scaled Jacobi matrices acquire the Laurent right limit with symbol \(z+z^{-1}\). The same symbol controls all polynomial fluctuations.

For the trace \(S_n=\sum_jX_j\), something stronger happens before taking any limit:

$$ \mathbb{E}S_n=0,\qquad \operatorname{Var}(S_n)=b_n^2. $$

The variance is exactly one boundary edge of the compressed tridiagonal Jacobi matrix. There is only one crossing from the first \(n\) basis vectors to the rest, and its weight is \(b_n\). Therefore

$$ \frac{S_n}{b_n}\Longrightarrow N(0,1). $$

Equivalently, the confluent Hankel determinant has a Gaussian window:

$$ \frac{H_n(u/b_n)}{H_n(0)}\longrightarrow e^{-u^2/2} $$

uniformly for \(u\) in compact real sets.

The particles and polynomial zeros also share the explicit Ullman law after division by \(b_n\), with density

$$ \frac1{2\pi}\operatorname{arcosh}\left(\frac2{|x|}\right) \mathbf1_{\{0<|x|<2\}}. $$

That macroscopic cloud has its own post. Here it is better viewed as one organ in the ensemble, not the whole creature.

The cost of the whole cloud#

The zero-time partition function satisfies

$$ \log H_n(0) =n^2\left( \log n-\log\log n+\log\pi-\frac32 \right)+o(n^2), $$

or

$$ H_n(0)^{1/n^2} \sim\frac{\pi n}{e^{3/2}\log n}. $$

This comes from the exact norm product and the weighted sum of \(\log b_j\). The Lambert denominator contributes \(-n^2\log\log n\); the triangular weighting of \(\log j\) contributes the slightly rude constant \(-3/2\). Local recurrence data have become thermodynamic-scale free energy.

Everything here is unconditional. It uses the positive real-axis weight \(1/\xi(1/2+x)\), plus a narrow zero-free complex neighborhood needed for the quantitative recurrence theorem. It does not assume the Riemann hypothesis.

And it cannot imply RH. Premise-matched counterfeit even entire functions can remain positive on the real axis, preserve the relevant convexity or log-Freud asymptotics, retain the same recurrence scale, CLT, particle law, free energy, or even the \(7/8\) correction, while carrying explicit zeros off the imaginary axis. The ensemble anatomy is real and rather beautiful. It simply does not remember enough of the complex zero divisor to become an RH criterion.

Notebook references: C-0027, C-0028, C-0031, K-0002