Investigating Special Values of Riemann Zeta Function at Even Arguments

This paper discusses the core properties and specific values of the Riemann zeta function, , which is central to analytic number theory and prime number distribution. This paper begins with the Basel Problem, a historic problem about the sum of the reciprocals of the squares of natural numbers. Leonhard Euler's neat solution to the Basel Problem revealed and opened the window of interactions from fundamental geometric constants to infinite series. After providing Euler's intuition, the paper explores the generalization of through many avenues, emphasizing the value of Bernoulli numbers for fluid closed-form expressions. The author fully verifies how power series expansions, contour integration, and Fourier series are all converging into the math leading people to the zeta-related identities. The author also discusses where the Riemann zeta function is used outside of pure math. In physics, especially quantum and statistical modeling, the author explores computational techniques. These are essential for advancing peoples understanding of prime number distribution and modern cryptography, while also observing connections in math from past to future.