Prime product formulas for the Riemann zeta function and related identities

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta (s)$ valid for $\mathrm{\Re }(s)>1$, as well as similar formulas for $\zeta (s)$ valid for an even and odd $k$th positive integer argument. We shall further give a set of generated formulas for $\zeta (k)$ up to $11$th order, including Ap'ery’s constant, and also construct formulas for $\zeta (3\mathrm{/}2)$. We’ll also validate these formulas numerically.