The Connection Between Prime Distribution and Random Matrices

The development of prime numbers has always been accompanied by humanitys pursuit of logical rigor and breakthroughs in abstract laws. The evolutionary trajectory of data science has always been closely linked to leaps in computing power and the explosive growth of data volume, providing robust support for in-depth explorations of prime number laws. In this paper, Python tools (e.g., NumPy) are used to acquire data on prime numbers and random matrices. Through normalization processing, comparison of statistical distributions, and visualization research, the statistical correlation between prime numbers (natural numbers greater than 1 that are divisible only by 1 and themselves) and random matrices (matrices whose elements are random numbers) is explored. The paper finds that there is a significant similarity between the distribution law of prime numbers (zeros of the Riemann zeta-function) and the distribution of eigenvalues of random matrices (Gaussian Unitary Ensemble (GUE) model). This statistical insight provides a new perspective for understanding the randomness and orderliness characteristics of prime number distribution in number theory and the application of random matrix theory in interdisciplinary mathematics.