A very short recursive formula for the $n$-th prime is (by a combination of Euclids proof and the sieve of Erathostenes):
$$p_{n+1} = \min_{x>1,\gcd(x,p_1\cdots p_n)=1} x$$
Using an approximation for the $\min$ function, we find:
$$p_{n+1} = \lim_{\rho \rightarrow \infty} -\frac{1}{\rho} \log(\sum_{x>1,\gcd(x,p_1\cdots p_n)=1} \exp(-\rho x))$$
My idea is to sum over all numbers $x>1$ but to weight them:
$$p_{n+1} =^? \lim_{\rho \rightarrow \infty} -\frac{1}{\rho} \log \left( \sum_{x\ge 2}^\infty \exp(-\rho x)\cdot \exp(-\gcd(p_1 \cdots p_n,x)^{\rho})\right )$$
Can this last equality be proven? (I have checked it numerically with SAGEMATH).
Is there any benefit in using this formula to deduce any property of the primes?
Thanks for your help.