When Graph Neural Networks Meet the Riemann Hypothesis: A Systematic Negative Study

Abstract

Graph Neural Networks have achieved remarkable success on graph prediction tasks across chemistry, social networks, and combinatorial optimization. But what happens when you point a GNN at one of the deepest open problems in mathematics? We conducted seven systematic experiment tracks testing whether GNNs can predict spectral properties of algebraic graphs whose eigenvalues are connected, through a chain of known theorems, to the Riemann zeta function and ultimately to the Riemann Hypothesis.

The answer is a definitive no. Across subgraph and full-graph architectures, Cayley graphs and Farey graphs and Pizer graphs, spectral gap prediction and Hecke eigenvalue prediction, every GNN configuration either fails outright or provides only marginal improvement over a trivial logarithmic baseline. The best result across all experiments is a ΔR² of +0.042 over linear regression. We trace this failure to a single theoretical root cause: vertex-transitivity, a structural property shared by Cayley graphs, makes every node look identical to a message-passing GNN, destroying the local structural diversity these networks depend on. We formalize this through the Weisfeiler-Leman hierarchy and prove that standard message-passing GNNs are limited to functions of diameter and degree on vertex-transitive graphs.

To our knowledge, no published work has applied GNNs to number-theoretic graph structures. These experiments constitute the first systematic investigation at this intersection, and they establish clear, hard limits.

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Introduction: Why GNNs and Number Theory?

The motivation behind this study starts with an enticing chain of mathematical connections. Consider a 4-regular graph built from the group SL(2, Fₚ), the group of 2×2 matrices with determinant 1 over the finite field with p elements. The eigenvalues of this graph's adjacency matrix encode expansion properties. Through the Lubotzky-Phillips-Sarnak construction [LPS, 1988] and Pizer's theorem [Pizer, 1990], those eigenvalues coincide with Hecke eigenvalues of modular forms. Those eigenvalues, in turn, control the analytic behavior of L-functions. And L-functions connect, through well-studied machinery, to the zeros of the Riemann zeta function ζ(s).

The chain looks like this:

Graph eigenvalues → Hecke eigenvalues → L-functions → ζ(s)

If a machine learning model could predict spectral properties of these graphs from their structure alone, it would open a computational pathway into territory that mathematicians have explored with purely symbolic methods for over a century. Graph Neural Networks seem like the natural tool for this job. They excel at learning relationships between local graph structure and global properties. Molecular property prediction, social network analysis, combinatorial optimization: GNNs dominate these domains by aggregating information from local neighborhoods.

We decided to test whether this promise extends to algebraic graphs connected to number theory. The answer, as it turns out, reveals more about the limitations of GNNs than about the Riemann Hypothesis.