Approximating the Riemann Zeta Function on the Critical Line using Neural Networks with Fourier Feature Embeddings

This research presents a deep learning surrogate model designed to numerically approximate the Riemann Zeta function, ζ(0.5+it)ζ(0.5+it), along the critical line. To overcome the low-frequency spectral bias typical of standard Multi-Layer Perceptrons (MLPs), the architecture incorporates a Fourier Feature Embedding layer, effectively allowing the network to learn the highly oscillatory behavior of the Zeta function. The model is trained on high-precision data from the mpmath library over the domain t∈[0,35]t∈[0,35], achieving a Mean Squared Error (MSE) of 1.76×10−61.76×10−6. Beyond interpolation, the network successfully identifies the first five non-trivial zeros of the function with a mean error of 3.8×10−33.8×10−3 in the tt-parameter. Finally, by exporting the model to the ONNX format and applying FP16 quantization, an ultra-low inference latency of just 0.0305 ms per evaluation is achieved (over 32,000 inferences per second on CPU), making it highly suitable for real-time mathematical applications.