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 Laurinv{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 )–( 55times ) 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) + lnzeta(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 ).