  A new derivation of Golomb's limit formula for generating the $n$th$+1$ prime
number is presented. The limit formula is derived by extracting $p_{n+1}$ from
Euler's prime product representation of the Riemann zeta function $\zeta(s)$ in
the limit as $s$ approaches infinity. Also, new variations of these limit
formulas are explored, such as the logarithm and a half-prime formulas for the
$p_{n+1}$.
