Orthogonal Polynomials of the Semilocal Euler–Gamma Measure: Effective Reconstruction of the Weil Form and an Arcsine Limit Theorem at the Prime Two

Abstract

For a finite set $\mathcal{P}$ of primes and $\sigma>0$ let $w_{\mathcal{P},\sigma}(t)=\lvert \mathcal{L}_{\mathcal{P}}(\sigma+it)\rvert^2$, where $\mathcal{L}_{\mathcal{P}}(s)=\pi^{-s/2}\Gamma(s/2)\prod_{p\in\mathcal{P}}(1-p^{-s})^{-1}$. At $\sigma=\tfrac{1}{2}$ this is the spectral measure of the semilocal scaling operator, and its logarithmic $\sigma$-derivative produces the archimedean and prime terms of the semilocal Weil quadratic form. We study the orthogonal polynomials of these measures from this viewpoint. First, for every $\lambda>1$ and every $\mathcal{P}$ containing the primes $p\le\lambda^2$, truncating the complete Weil form in the polynomial grading reconstructs it with an explicit error $O(\log N/N)$, uniformly on the unit ball of $H^2$ functions supported in $[-\log\lambda,\log\lambda]$. Second, for $\mathcal{P}=\{2\}$ and $\sigma\in[\tfrac{1}{4},\tfrac{3}{4}]$, we prove that the normalized inverse Gram corner $d_n(\sigma)$, which governs the ratios of the Jacobi coefficients of $w_{\{2\},\sigma}$ and $w_{\varnothing,\sigma}$, converges in $C^1$ as $n\to\infty$. The limit is a series of ordered words in the coordinate projection onto $\ell^2(\mathbb{Z}_{\le0})$ and spectral projections of the operators $\cosh(2b)-\sinh(2b)X$, $b\ne0$, on $\ell^2(\mathbb{Z})$, whose spectral measures are arcsine laws; already its second-order term differs from the value predicted by Toeplitz theory. Consequently the ratios of the Jacobi coefficients and of the leading coefficients converge, and the $\sigma$-derivative of $\log a_n(\{2\},\sigma)-\log a_n(\varnothing,\sigma)$ tends to zero uniformly in $\sigma$. The proofs exploit that multiplication by $e^{iut}$ acts in the Meixner–Pollaczek basis as a hyperbolic element of $SU(1,1)$. Time flies, and I want to explore more of the unknown.