Perspectives on the arithmetic nature of the ratios $\zeta (2n+1)\mathrm{/}{\pi }^{2n+1}$ and $\beta (2n)\mathrm{/}{\pi }^{2n}$

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective. Beyond analytic interest, these formulas motivate linear independence conjectures which, if established, would imply the irrationality of the quantities $\zeta (2n+1)\mathrm{/}{\pi }^{2n+1}$ and $\beta (2n)\mathrm{/}{\pi }^{2n}$