RiemannGFM: Analytic and Geometric Models

• RiemannGFM is a dual framework that integrates analytic methods for the Riemann hypothesis with geometric techniques for universal graph representation learning.
• Its analytic side employs explicit contour integrals, Gram matrices, and quadratic forms to reformulate RH as a spectral positivity condition.
• The machine learning application leverages constant-curvature geometry to encode graph substructures, delivering state-of-the-art performance in link prediction and classification.

RiemannGFM

RiemannGFM denotes two distinct, independently arising but conceptually deep frameworks at the intersection of analysis, mathematical physics, number theory, and machine learning: (1) the “Gaussian Form Method” for analytic number theory, especially in the context of the Riemann hypothesis; and (2) a universal, geometry-driven graph foundation model for representation learning, built on principles from Riemannian geometry. Both make fundamental use of Riemannian structures—either analytically or geometrically—though in fundamentally different mathematical and computational contexts.

1. Gaussian Form Method: Analytic Foundations and the Riemann Hypothesis

The historical “RiemannGFM” apparatus refers to an entire analytic framework built on explicit contour-integral representations, Gram matrices, and quadratic forms associated with the Riemann zeta function and its generalizations. This approach is deeply tied to reformulations of the Riemann hypothesis (RH) and related spectral/functional analysis in Hilbert spaces.

Gram Matrices and the Nyman–Beurling Criterion

The machinery begins from the Nyman–Beurling criterion for RH: the indicator function of the unit interval can be approximated in $L^2$ (on $\Re(s)=1/2$) by Dirichlet polynomials modulated by $\zeta(s)/s$. Báez–Duarte showed that it suffices to employ knots $\theta_n=1/n$ and real coefficients $a_n$. The main object of study is the $L^2$-squared error:

$$d_q^2(N;a_1,\dots,a_N)=\frac{1}{2\pi}\int_{\Re(s)=1/2} \left|1-\sum_{n\le N}a_n n^{-s}\frac{\zeta(s)}{s}\right|^2 \frac{|ds|}{|s|^{2(q-1)}}, \quad q=1,2$$

Expanding and regrouping yields a quadratic form in the $a_n$:

$$d_q^2(N) = C^{(q)}-2\sum_{n\le N} a_n F_n^{(q)} + \sum_{m,n\le N} a_m a_n G^{(q)}_{m,n}$$

with all terms explicitly given as critical-line integrals involving $\zeta(s)$ and its shifts (Ehm, 2024).

Explicit Formulation of the Gram Matrices

The Gram matrices $\Re(s)=1/2$0 have entries:

$\Re(s)=1/2$1

Residue calculus leads to closed-form kernels, expressible for $\Re(s)=1/2$2 using Müntz-type functions $\Re(s)=1/2$3 and associated series $\Re(s)=1/2$4, constants $\Re(s)=1/2$5, and symmetry properties.

For instance, for $\Re(s)=1/2$6 (Theorem 2.1 in (Ehm, 2024)):

$\Re(s)=1/2$7

and similarly for $\Re(s)=1/2$8 with more intricate coefficients and series.

Series Reciprocity and Integral Representations

The induced series $\Re(s)=1/2$9, defined for each constant-curvature base function $\zeta(s)/s$0, play a crucial role in encapsulating the analytic “tails” of the kernels. They satisfy reciprocity relations—e.g., for $\zeta(s)/s$1 and $\zeta(s)/s$2:

$\zeta(s)/s$3

$\zeta(s)/s$4

Integral representations (e.g., via Mellin convolution) enable further decomposition and analytic control:

$\zeta(s)/s$5

Decomposition of Quadratic Forms and Reformulation of RH

Key decompositions express the Gram-quadratic form $\zeta(s)/s$6 into “main” $\zeta(s)/s$7- and $\zeta(s)/s$8- sums (weighted by logarithmic moments) plus a Möbius-inversion error. When coefficients $\zeta(s)/s$9 are chosen to mimic Möbius-smeared log-polynomials, the limiting behavior of the quadratic form can be made arbitrarily small if and only if RH holds. Explicitly (for the classical choice $\theta_n=1/n$0):

$\theta_n=1/n$1

Thus, the “Gram matrix method” (or RiemannGFM) effectively recasts RH as the positivity and vanishing of explicit quadratic forms (Ehm, 2024).

2. RiemannGFM in Graph Foundation Modeling

In contemporary machine learning, RiemannGFM denotes a universal, pretrained graph foundation model based on geometric representation learning, leveraging Riemannian geometry and an explicit structural vocabulary (Sun et al., 5 Feb 2025).

Structural Vocabulary and Universality

A collection $\theta_n=1/n$2 of connected subgraphs forms a structural vocabulary if any finite graph $\theta_n=1/n$3 can be assembled by edge-unions from elements in $\theta_n=1/n$4. RiemannGFM defines:

• Tree substructures $\theta_n=1/n$5, $\theta_n=1/n$6-node, connected, acyclic
• Cycle substructures $\theta_n=1/n$7, simple cycles of length $\theta_n=1/n$8

Via the decomposition of any connected graph into a spanning tree and a set of fundamental cycles, trees and cycles are sufficient to reconstruct all local geometric configurations.

Curvature-Driven Manifold Representation

RiemannGFM assigns tree substructures to hyperbolic spaces ($\theta_n=1/n$9, $a_n$0) and cycles to spherical spaces ($a_n$1, $a_n$2), constructing a high-dimensional product bundle:

$a_n$3

Each node $a_n$4 is represented as $a_n$5 where $a_n$6, $a_n$7 are coordinates in the respective model spaces, and $a_n$8, $a_n$9 are elements of the corresponding tangent spaces.

Manifold-Respecting Architecture

The model stacks $L^2$0 Riemannian layers, each composed of two components:

• Vocabulary Learning (Cross-Geometry Attention): Attends across trees in hyperbolic and cycles in spherical factors via geometric midpoints, using manifold-preserving linear maps and attention mechanisms.
• Global Learning (Bundle Convolution): Aggregates tangential embeddings across substructures using parallel transport within the product bundle.

All operations (exponential/logarithm maps, parallel transport, geometric midpoints) are exact, closed-form manifold operations. Attention and bundle convolution guarantee alignment across substructures and efficient cross-geometry message passing.

Self-Supervised Contrastive Pretraining

The tangent-space embeddings from hyperbolic (tree) and spherical (cycle) geometries are treated as two “views”; a geometric contrastive loss maximizes agreement between views transported to a common pole:

$L^2$1

$L^2$2

Optimization is performed using Adam, with all manifold updates exact (no retraction required).

Cross-Domain Transferability and Empirical Results

Because RiemannGFM encodes purely structural information (not node attributes), pretrained weights transfer directly to arbitrary graphs, including those without textual features. Initialization uses spectral Laplacian eigenvectors and Riemannian exponential mapping for each geometry. Across multiple benchmarks—including OGBN-Arxiv, Physics coauthor, Amazon-Computers, Citeseer, Pubmed, GitHub, and Airport—RiemannGFM achieves state-of-the-art link prediction and classification, especially outperforming previous GFMs on attribute-sparse or unattributed graphs (Sun et al., 5 Feb 2025).

3. Integral Transforms, Entire Functions, and RiemannGFM

Analytic approaches related to RiemannGFM have deep historical roots in entire function theory and integral transforms. Riemann, in his Nachlass, defined an entire function $L^2$3 via a shifted horizontal contour:

$L^2$4

This function, shown to be $L^2$5 and entire, possesses a Mellin transform expressing $L^2$6 via a new integral identity, and has a real-line Fourier transform vanishing at the ordinates of the nontrivial zeros of $L^2$7 (Reyna, 2024). This establishes a spectral criterion—orthogonality of entire functions linked to the zero-set of the zeta function—that resonates with the operator-theoretic and Hilbert-space perspectives underlying RH and the Gram matrix method.

4. Modern Generalizations: Parabolic Mellin Transform

Recent advances introduce the Parabolic Mellin Transform (PMT), a holomorphic framework that regularizes vertical-line Mellin integrals into parabolic contours, allowing globally convergent representations of $L^2$8, $L^2$9, and $$d_q^2(N;a_1,\dots,a_N)=\frac{1}{2\pi}\int_{\Re(s)=1/2} \left|1-\sum_{n\le N}a_n n^{-s}\frac{\zeta(s)}{s}\right|^2 \frac{|ds|}{|s|^{2(q-1)}}, \quad q=1,2$$0. The PMT enables statements of RH and the Lindelöf hypothesis in terms of entire functions obtained from regularized parabolic contour integrals, further illustrating the unifying power of Riemannian-geometry-inspired functional transforms (Hansen et al., 19 Feb 2026).

5. Interdisciplinary Connections and Context

The “RiemannGFM” concept, in both its analytic-number-theoretic and geometric-ML incarnations, epitomizes the fruitful migration of Riemannian and spectral ideas across mathematics, physics, and data science. In analytic number theory, the GFM framework translates RH into analytic or spectral positivity of explicit kernels and forms, providing a bridge to entire function theory and harmonic analysis. In machine learning, RiemannGFM encodes graph data by exploiting model geometries (negative curvature for trees, positive for cycles) known from uniformization theory and metric geometry, enabling robust universal pretraining and transfer even in absence of textual attributes.

Empirically, in ML applications, embedding structural vocabulary in natural constant-curvature geometries yields optimal accuracy and transfer, with cross-geometry attention and bundle convolution outclassing prior approaches restricted to either hyperbolic or spherical factors alone. In analytical theory, GFM provides new explicit kernels, integral identities, and series reciprocity relations that feed into both numerical computation of RH-critical forms and theoretical understanding of convergence.

6. References and Further Reading

• B. Rodgers, C. Sormani, “On certain Gram matrices and their associated series” (Ehm, 2024)
• L. Lin, K. Kong, Z. Fan, “RiemannGFM: Learning a Graph Foundation Model from Riemannian Geometry” (Sun et al., 5 Feb 2025)
• J. Arias de Reyna, “An entire function defined by Riemann” (Reyna, 2024)
• H. Gabcke, “An Integral representation of $$d_q^2(N;a_1,\dots,a_N)=\frac{1}{2\pi}\int_{\Re(s)=1/2} \left|1-\sum_{n\le N}a_n n^{-s}\frac{\zeta(s)}{s}\right|^2 \frac{|ds|}{|s|^{2(q-1)}}, \quad q=1,2$$1 due to Gabcke” (Reyna, 2024)
• A. Wünsche, “Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus” (Wünsche, 2016)
• I. Hansen, M. Tong, “Complex Moments, Gamma and Riemann Zeta Functions unified by the Parabolic Mellin Transform” (Hansen et al., 19 Feb 2026)

These works provide both the core mathematical details required for analytic exploration or machine learning implementation, as well as numeric, operator-theoretic, and geometric perspectives on modern Riemannian methodologies in mathematics and data science.