In this talk, we explore the deep connection between Bernoulli numbers ($B_n$) and the special values of the Riemann Zeta function $\zeta(s)$. Starting from the classical problem of summing powers of integers, we revisit Euler's celebrated solution to the Basel Problem ($\sum 1/n^2 = \pi^2/6$).
Using tools from complex analysis—specifically infinite product expansions and the analytic continuation of $\zeta(s)$—we derive the exact closed-form expression for $\zeta(2n)$ in terms of Bernoulli numbers:
$$ \zeta(2n) = \frac{(-1)^{n-1} 2^{2n-1} B_{2n}}{(2n)!} \pi^{2n} $$
Finally, we discuss the functional equation relating $\zeta(s)$ to $\zeta(1-s)$ and briefly touch upon the mysterious nature of the odd values $\zeta(2n+1)$.