Related papers: The Riemann $\Xi$-function from primitive Markovia…

In this paper, we introduce the spectral projection operators $\mathbb{P}_m$ on non-degenerate nilpotent Lie groups $\mathcal{N}$ of step two, associated to the joint spectrum of sub-Laplacian and derivatives in step two. We construct their…

This paper studies the integral of the Riemann xi-function. More generally, it studies a one-parameter family of functions given by Fourier integrals and satisfying a functional equation. Members of this family are shown to have only…

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincar\'e metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $\Omega^{\otimes k}_X$, where $\O$ is the…

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which…

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to…

An element [\Phi] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis {\phi} for C(x). The Pl\"ucker coordinates…

Let $\pi$ be a Speh module of $\mathbf{GL}(2n,\mathbb{R})$ based on a discrete series of $\textbf{GL}(2,\mathbb{R})$. The aim of this paper is to build a chain complex of $\pi$ by direct sum of auto-duals standard modules,…

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical…

We present a kernel-based methodology for constructing Lyapunov functions for nonlinear dynamical systems using approximate Koopman eigenfunctions. Our approach decomposes principal Koopman eigenfunctions into linear and nonlinear…

We introduce a polyanalytic extension of the Gaussian radial basis function (RBF) kernel by computing the action of the convolution operator on normalized Hermite functions. In particular, using the Zaremba-Bergman formula we derive an…

We initiate the study of spectral zeta functions $\zeta_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions.…

The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library,…

We obtain complementary recurrence and transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi_{n+1}$. Here $M$ denotes a primitive matrix having…

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other $L$-functions. We establish…

Starting from the classical integral representation of the $\zeta(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we…

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel…

The Riemann zeta function and the distribution of its nontrivial zeros on the critical line remain central topics in analytic number theory and large-scale computation. This work develops a numerical framework that replaces classical…

We identify a structural property of term-rewriting proof systems called operational inexpressibility: no derivation depends on a specified input dimension and also constrains the target question. The canonical instance is direct…

We introduce from an analytic perspective Christoffel-Darboux kernels associated to bounded, tracial noncommutative distributions. We show that properly normalized traces, respectively norms, of evaluations of such kernels on finite…

We consider Riemann's Xi function $\xi(s)$ which is evaluated at $s = \frac{1}{2} + \sigma + i \omega$, given by $\xi(\frac{1}{2} + \sigma + i \omega)= E_{p\omega}(\omega)$, where $\sigma, \omega$ are real and compute its inverse Fourier…