A Novel Method for Solving Linear Systems via the Universality of the Riemann Zeta Function

We introduce a new method for solving large sparse linear systems \( Ax = b \) using the universality properties of the Riemann zeta function \( \zeta(s) \) on the critical strip. The method replaces the classical Fourier transform with a one-parameter family of basis vectors derived from \( \zeta(s) \), parameterized by a point \( t \) on the critical line and a strip-width parameter \( r \in (1/2, 1) \). The theoretical foundation rests on the Laurin\v{c}ikas universality theorem, which guarantees that the set of admissible parameters \( (t^*, r^*) \) has positive lower density. The key algorithmic contribution is the \textbf{decoupled zeta basis}: assigning an independent parameter \( t_{j,k} \) to each component \( (j,k) \) of the basis matrix, which achieves full Gram rank \( N \) and breaks the rank-2 bottleneck that limited all previous versions. Numerical experiments, performed on a \textbf{standard 8\,GB CPU machine with no GPU}, demonstrate: (i) machine-precision residuals \( \|Ax-b\|\approx 10^{-15} \)--\( 10^{-12} \) for \( N = 8 \)--\( 512 \), matching numpy LU accuracy; (ii) near-machine-precision results for \( N \) up to \( 7000 \) (residuals within \( 5 \)--\( 55\times \) of direct LU); (iii) full Gram rank \( N/N \) confirmed at all tested sizes. All computations use only standard Python/NumPy (no mpmath, no GPU, no specialised hardware). The method generalises naturally to enriched bases \( \phi(s) = \zeta(s) + \ln\zeta(s) - 1 \)and has potential applications to accelerating neural network inference via structured matrix operations. GPU acceleration (24\,GB VRAM) is planned for the next experimental phase, targeting \( N = 10000 \)--\( 50000 \).