--- title: Stochastic Zeta Function url: https://www.emergentmind.com/topics/stochastic-zeta-function type: topic --- # Stochastic Zeta Function Searching arXiv for recent and foundational papers on “stochastic zeta function” to ground the article. The term *stochastic zeta function* is used in several mathematically distinct senses. In random matrix theory and analytic number theory, it most commonly denotes a random entire function whose zeros are given by a limiting point process such as the sine-kernel or $\mathsf{Sine}_\beta$ process, arising as the microscopic limit of rescaled characteristic polynomials and conjecturally of suitable ratios of the Riemann zeta function [2202.04284], [2009.04670]. In topological probability, the same term refers to Mellin transforms of persistence-bar counts associated with superlevel sets of stochastic processes, yielding meromorphic objects with poles determined by small-scale oscillation statistics [2110.10982]. Other uses include probabilistic Mellin-transform representations of completed zeta functions [2411.08863] and kernels built from periodic zeta functions for Gaussian process regression [2208.02596]. The phrase therefore names a family of constructions rather than a single invariant, but across these settings it consistently links zeta-type analytic structure to stochastic data. ## 1. Random entire functions with random zero sets In the random-matrix-theoretic usage, the stochastic zeta function is a random holomorphic function defined by its zeros. For the circular unitary ensemble, if \[ Z_n(X)=\det(\mathrm{Id}-U_n^*X), \] then the microscopic rescaling near $X=1$ is \[ \xi_n(s)=\frac{Z_n(e^{2 i \pi s/n})}{Z_n(1)}, \qquad s\in\mathbb{C}. \] Its zeros are the centered and rescaled eigenangles, and these zeros converge to the determinantal sine-kernel point process on $\mathbb{R}$ [2202.04284]. The limiting function is \[ \xi_\infty(s)=e^{i\pi s}\prod_{k\in\mathbb{Z}}\Bigl(1-\frac{s}{y_k}\Bigr), \] with the infinite product understood in the principal value sense, where $(y_k)_{k\in\mathbb{Z}}$ are the points of the sine-kernel process [2202.04284]. In this formulation, the stochastic zeta function is almost surely entire, of order $1$, normalized by $\xi_\infty(0)=1$, and has simple real zeros distributed according to the limiting point process [2202.04284]. A broader construction replaces the sine-kernel process by a simple locally finite point process $X$ on $\mathbb{R}$ with counting measure \[ M_X=\sum_{\rho\in X}\delta_\rho. \] Under suitable integrability, growth, and tightness assumptions, one defines \[ f(s)=\lim_{A\to\infty}\exp\!\left(s\int_{[-A,A]}\frac{dM_0(\rho)}{\rho}\right)\prod_{\rho\in X\cap[-A,A]}\Bigl(1-\frac{s}{\rho}\Bigr), \] where $M_0$ is an appropriate deterministic Radon measure [2202.04284]. Proposition 2.4 in that work identifies this principal-value construction with a genus-$1$ Weierstrass product involving \[ E_1(u)=(1-u)e^u, \] thereby placing the function in a canonical Hadamard-product framework [2202.04284]. A distinct but closely related formulation appears in the $\mathsf{Sine}_\beta$ setting. There, the random entire function $\zeta_\beta$ is defined as the unique law on Cartwright-class entire functions satisfying $\zeta_\beta(0)=1$, $\zeta_\beta(\mathbb{R})\subset\mathbb{R}$, and having zeros given by the $\mathsf{Sine}_\beta$ process [2009.04670]. Equivalent representations include a principal value infinite product over the $\mathsf{Sine}_\beta$ points, a Brownian-motion power series, a regularized determinant of the $\mathsf{Sine}_\beta$ operator, and an SDE-based representation [2009.04670]. This makes the stochastic zeta function simultaneously an entire function, a spectral determinant, and a stochastic differential object. ## 2. Point-process convergence, canonical products, and universality A central structural result is that convergence of point processes can imply convergence of the associated random holomorphic functions. If $(X_n)$ are simple locally finite point processes on $\mathbb{R}$ converging vaguely in law to $X$, and if deterministic comparison measures $M_{0,n}$ and $M_0$ satisfy the stated tail integrability, approximation, and tightness hypotheses, then the associated canonical products \[ f_n(s)=\lim_{A\to\infty}\exp\!\left(s\int_{[-A,A]}\frac{dM_{0,n}(\rho)}{\rho}\right)\prod_{\rho\in X_n\cap[-A,A]}\Bigl(1-\frac{s}{\rho}\Bigr) \] converge in law, for the topology of uniform convergence on compact sets, to the limiting entire function $f$ defined analogously from $X$ and $M_0$ [2202.04284]. This theorem provides a unified mechanism for passing from spectral point-process limits to holomorphic-function limits. That framework covers multiple classical ensembles. For rotationally invariant circular ensembles, the rescaled eigenangle process converges to a simple limiting process, equal to $\mathsf{Sine}_\beta$ for $\mathrm{C}\beta\mathrm{E}$, and the corresponding products \[ \xi_n(s)=e^{i\pi s}\prod_{\rho\in X_n}\Bigl(1-\frac{s}{\rho}\Bigr) \] converge in law ucp to \[ \xi_\infty(s)=e^{i\pi s}\prod_{\rho\in X}\Bigl(1-\frac{s}{\rho}\Bigr) \] [2202.04284]. In the Gaussian ensembles, bulk rescalings again converge to $\mathsf{Sine}_\beta$, and the characteristic polynomial ratios converge to stochastic-zeta-type limits with explicit deterministic exponential normalizations depending on the energy parameter $E$ [2202.04284]. At the soft edge, analogous canonical products are built from the Airy point process [2202.04284]. The 2026 work on Hua–Pickrell stochastic zeta functions extends this perspective. For $\beta>0$ and $\delta>0$, the Hua–Pickrell point process $\mathcal{HP}_{\beta,\delta}$ is defined through a coupled SDE family, and the associated stochastic zeta function is the principal-value product \[ \boldsymbol{\xi}^{\beta,\delta}(z):=\lim_{R\to\infty}\prod_{\substack{\mathsf{x}\in\mathcal{HP}_{\beta,\delta}\\ |\mathsf{x}|0$ [2009.04670]. The same work proves uniqueness of the law in the Cartwright class under the conditions $\zeta_\beta(0)=1$, $\zeta_\beta(\mathbb{R})\subset\mathbb{R}$, and zero set equal to $\mathsf{Sine}_\beta$ [2009.04670]. The Hua–Pickrell extension preserves this operator-theoretic viewpoint. Assiotis and Najnudel state that Li–Valkó introduced $\tilde{\boldsymbol{\xi}}^{\beta,\delta}$ as the secular function of a random Dirac operator whose spectrum is $\mathcal{HP}_{\beta,\delta}$, and that the principal-value product definition is distributionally equivalent to this operator-theoretic construction [2602.08739]. The logarithmic derivative again takes the form of a principal-value sum over spectral points [2602.08739]. A plausible implication is that stochastic zeta functions in the random-operator setting are best understood as random spectral determinants with entire-function regularity encoded by Cartwright or Hadamard theory. ## 4. Relations to the Riemann zeta function and mesoscopic random models One major motivation for the terminology is the analogy with the Riemann zeta function itself. In the point-process framework, the nontrivial zeros of $\zeta(s)$ are rescaled near a height $t_0$ by the microscopic scale $2\pi/\log t_0$, and ratios of zeta values are expressed in terms of products over nearby zeros [2202.04284]. The stochastic zeta conjecture quoted there states that, if $\omega\sim \mathrm{Unif}[0,1]$ and $T\to\infty$, then \[ \left(\frac{\zeta\bigl(\tfrac12+i\omega T-\tfrac{i2\pi s}{\log T}\bigr)}{\zeta(\tfrac12+i\omega T)}\right)_{s\in\mathbb{C}} \Rightarrow \bigl(\xi_\infty(s)\bigr)_{s\in\mathbb{C}} \] in law ucp [2202.04284]. Theorem 1.1 in that paper proves an equivalence between this conjecture and the GUE conjecture under the Riemann Hypothesis and simplicity of zeros [2202.04284]. The same work establishes a Cauchy limit law for the logarithmic derivative at typical heights: \[ \left(\frac{2i}{\log T}\frac{\zeta'}{\zeta}\!\left(\tfrac12+i\omega T\right)+i\right)\Rightarrow \text{standard Cauchy} \] [2202.04284]. This parallels the Cauchy-law universality for Stieltjes transforms of translation-invariant random atomic measures proved in the same paper [2202.04284]. A different stochastic theory of the Riemann zeta function appears in work on Gaussian multiplicative chaos. Saksman and Webb prove that \[ x\mapsto \zeta\!\left(\tfrac12+i\omega T+ix\right) \] converges, after insertion of a mild decay factor, in law in negative Sobolev space to a non-trivial random generalized function [1609.00027]. They identify the limit as a product $g(x)v(x)$, where $g$ is a random smooth nonvanishing function and $v$ is a complex Gaussian multiplicative chaos distribution with covariance kernel involving $\log \zeta(1+i(x-y))$ [1609.00027]. On mesoscopic scales, they show that $\zeta(1/2+i\delta_T x+i\omega T)$ decomposes asymptotically into a divergent scalar factor predicted by Selberg’s central limit theorem, a smooth factor approaching $1$, and a strict complex Gaussian multiplicative chaos distribution [1609.00027]. Related stochastic approximations of the Euler product produce multiplicative-chaos measures for a random model of the zeta function. In that model, \[ X_N(x)=\sum_{j=1}^N p_j^{-1/2}\cos(\theta_{p_j}-x\log p_j) \] is a randomized prime sum, and the measures \[ \mu_{\beta,N}(dx)=\frac{e^{\beta X_N(x)}}{\mathbb E[e^{\beta X_N(x)}]}\,dx \] converge in the subcritical regime to a non-Gaussian multiplicative chaos measure and at criticality after Seneta–Heyde normalization [1604.08378]. These are not stochastic zeta functions in the entire-function sense, but they belong to the same probabilistic program of modeling zeta through log-correlated random structures [1604.08378]. The term also appears in a more heuristic prime-side setting. França and LeClair describe a “stochastic zeta function” viewpoint in which random-walk-like cancellation in prime sums supports a truncated Euler-product approximation and a probabilistic zero model for $\zeta(s)$ [1601.00914]. That work labels key claims as “Proposal,” so its conclusions are more conjectural than the random-matrix and multiplicative-chaos constructions [1601.00914]. ## 5. Topological and Mellin-transform zeta functions of stochastic processes A separate research line uses *stochastic zeta functions* for analytic transforms of topological summaries of sample paths. For a stochastic process $X$ on a compact interval, one considers the superlevel-set filtration $S_u=\{\tau: X(\tau)\ge u\}$ and its $0$-dimensional persistence diagram [2110.10982]. The $p$-persistence functional is \[ \mathrm{Pers}^p(\mathrm{Dgm}(X))=\sum_{(b_i,d_i)\in \mathrm{Dgm}(X)} (d_i-b_i)^p, \] with the infinite bar contributing the range of $X$ [2110.10982]. The corresponding global stochastic zeta function is \[ \zeta_X(p):=\mathbb E[\mathrm{Pers}^p(X)] = p\int_0^\infty \varepsilon^{p-1}\,\mathbb E[N_\varepsilon(X)]\,d\varepsilon, \] where $N_\varepsilon(X)$ counts persistence bars of length at least $\varepsilon$ [2110.10982]. A local version at level $x$ is defined analogously by counting diagram points in the rectangle $]-\infty,x]\times[x+\varepsilon,\infty[$ [2110.10982]. For $\alpha$-stable Lévy processes, the range term diverges in moments of order at least $\alpha$, so the paper introduces the tail zeta function \[ \hat{\zeta}_X(p):=\mathbb E[\mathrm{Pers}^p(X)-(\mathrm{range}(X))^p]. \] Its main theorem states that \[ \hat{\zeta}_X(p)=\frac{t^{p/\alpha}}{\Gamma(p/\alpha)}\,B^*(p/\alpha), \] and that this function admits a meromorphic extension to $\mathbb C$ with a single simple pole at $p=\alpha$, of residue $t/\mathbb E[U]$, where $U$ is the renewal inter-oscillation time at unit scale [2110.10982]. The pole reflects the asymptotic behavior $\mathbb E[N_\varepsilon(X)]\asymp \varepsilon^{-\alpha}$ [2110.10982]. In this literature, the analogy with classical zeta functions is explicitly partial. The paper states that these objects admit meromorphic continuation and residues analogous to number-theoretic zetas, but that there is no Euler product and no general functional equation outside special cases [2110.10982]. This makes the terminology structurally motivated rather than arithmetically literal. ## 6. Other meanings, applications, and terminological scope The term has also been used for probabilistic Mellin-transform realizations of classical zeta functions. For $K=\mathbb Q$, $\mathbb Q(i)$, or $\mathbb Q(\sqrt{-2})$, an explicit probability density $\phi_K$ on $(0,\infty)$ is constructed from Tate-style zeta integrals so that \[ \mathbb E[X^s]=|D|^{-s/2}\,\xi_K(s), \] where $\xi_K$ is the completed Dedekind zeta function and $X$ is a positive random variable with density $\phi_K$ [2411.08863]. In this setting, “stochastic zeta function” means that the completed zeta function is realized as the Mellin transform of a probability law [2411.08863]. The paper uses this interpretation to prove positivity of the first two Li coefficients for those fields [2411.08863]. Another distinct usage appears in probability laws derived from hyperbolic secant random variables. There, moments of a hyperbolic secant variable and of the sum of two independent such variables yield probabilistic derivations of formulas for $\beta(2n+1)$ and $\zeta(2n+2)$ in terms of Euler numbers [2410.05895]. This is zeta-related stochastic analysis rather than a canonical random holomorphic function, but it fits the broader pattern of encoding zeta values in random-variable transforms. Outside number theory and random matrices, the phrase can also refer to covariance kernels built from the periodic zeta function. The periodic covariance \[ Z_\nu(x)=\frac{\Re F(x,1+2\nu)}{\zeta(1+2\nu)} \] defines a stationary periodic Gaussian-process kernel with power-law Fourier spectrum and smoothness parameter $\nu$ analogous to the Matérn class [2208.02596]. This is called a “stochastic zeta function” covariance in that context, but the object is a covariance kernel rather than an entire random function [2208.02596]. These usages show that the phrase is now genuinely polysemous. The most established meaning in current arXiv literature remains the random entire function associated with sine-type point processes and microscopic characteristic-polynomial limits [2202.04284], [2009.04670], [2602.08739]. A common misconception is therefore to treat every stochastic zeta function as a random analogue of the Riemann zeta function in the same formal category. The literature instead contains at least four different categories: random entire functions with random zeros, Mellin-transform zeta laws for persistence statistics, probabilistic Mellin representations of classical zeta functions, and zeta-based stochastic kernels. ## 7. Significance and current directions Within random matrix theory, the stochastic zeta function has become a canonical bulk limit object. It organizes the passage from eigenvalue point processes to rescaled characteristic polynomials, supports operator-theoretic interpretations, and controls new formulas for $\mathsf{Sine}_\beta$ correlation functions and supercritical moments of the $\mathrm{C}\beta\mathrm{E}$ field [2602.08739]. In particular, Assiotis and Najnudel prove that all $m$-point correlation functions of $\mathsf{Sine}_\beta$ can be written through expectations of products of $|\boldsymbol{\xi}^{\beta,\delta}|$ [2602.08739]. They also prove a conjecture of Fyodorov and Keating on supercritical moments by expressing the asymptotic constant through moments of integrals of $|\boldsymbol{\xi}^{\beta,\delta}|^{2s}$ [2602.08739]. In analytic number theory, the stochastic-zeta viewpoint sharpens the analogy between local zeta statistics and random matrix limits. The equivalence between the stochastic zeta conjecture and the GUE conjecture, under the Riemann Hypothesis and simplicity of zeros, makes the entire-function limit formulation part of the standard conjectural dictionary between zeros of $\zeta$ and sine-kernel statistics [2202.04284]. At a different scale, Gaussian-field limits for mesoscopic zero statistics provide a rigorous stochastic description of fluctuations in terms of $H^{1/2}$ covariance structures [1203.5328]. In probability and topological data analysis, the persistence-based stochastic zeta functions show that zeta-type analytic continuation can encode pathwise oscillation geometry, renewal structure, and parameter estimation for Lévy and semimartingale models [2110.10982]. This suggests that “zeta function” techniques can be exported beyond arithmetic and spectral problems when Mellin transforms of scale-counting functionals have sufficiently rigid asymptotics. A plausible implication is that future usage of the term will continue to bifurcate: one branch centered on random entire spectral determinants and universality, the other on stochastic transforms whose poles, residues, or covariance structure summarize random geometric or probabilistic data. The existing literature already supports both trajectories. Source: https://www.emergentmind.com/topics/stochastic-zeta-function