In this talk, we introduce the study of Bernoulli numbers starting from sums of integer powers and their relationship with the generating function via $\coth(z)$. We present Euler's argument, which utilizes the infinite product representation of the sine function to establish the celebrated identity: $$\zeta(2n)=\frac{(-1)^{n-1}2^{2n-1}B_{2n}}{(2n)!}\pi^{2n}$$

Subsequently, we explore a modern probabilistic approach. We define the logarithmic variable $\Lambda:=\log|\mathbb{C}_1\mathbb{C}_2|$, where $\mathbb{C}_1$ and $\mathbb{C}_2$ are independent Cauchy variables. By computing the mathematical expectation $\mathbb{E}[\Lambda^{2n}]$ through direct analytical integration, the result is expressed in terms of the Riemann zeta function and the Gamma function. Alternatively, using the moment generating function, this same expectation is linked to the Secant (or Euler) numbers $L_{2n}$: $$\mathbb{E}[\Lambda^{2n}]=\left(\frac{\pi}{2}\right)^{2n}L_{2n}$$ Equating both methods yields an alternative expression for $\zeta(2n+2)$.

Finally, the presentation examines the landscape of the function evaluated at odd integers, $\zeta(2n+1)$. Euler proposed a conjecture regarding these values. The talk notes that in 1978, Roger Apéry gave a proof that $\zeta(3)$ is an irrational number. Furthermore, mathematicians such as Rivoal (2000) and Zudilin (2001) demonstrated the irrationality of other odd zeta values. We conclude by illustrating how the sum of the fractional parts of the zeta function values equals $1$, a result deeply connected to the probabilistic distribution of the reciprocal of a uniform random variable.