PRIME NUMBERS, ZETA FUNCTIONS, AND THE RIEMANN HYPOTHESIS: A COMPREHENSIVE REVIEW

Abstract

This comprehensive review examines the intricate relationship between prime numbers, the Riemann zeta function, and the celebrated Riemann Hypothesis. We analyze historical developments, current theoretical frameworks, and computational approaches in understanding prime number distribution. The study synthesizes classical results with modern computational methods, providing insights into one of mathematics' most profound unsolved problems. Special attention is given to recent algorithmic approaches and numerical verifications of the hypothesis up to significant heights on the critical line.