It is possible, by means of zeta function regularization and the Ramanujan summation method, to assign a finite value to the sum of the natural numbers (here $n \to \infty $) :
$$ 1 + 2 + 3 + 4 + \cdots + n \; {“ \;=\; ”} - \frac{1}{12} . $$
Is it also possible to assign a value to the sum of primes, $$ 2 + 3 + 5 + 7 + 11 + \cdots + p_{n} $$ ($n \to \infty$) by using any summation method for divergent series?
This question is inspired by a question on quora.
Thanks in advance,
Fröberg shows in his paper that the prime zeta function
$$P(s)=\sum_{p\in \mathbb P} \frac1{p^s}=\sum_{k=1}^\infty \frac{\mu(k)}{k}\log\zeta(ks)$$
where $\mu(k)$ and $\zeta(s)$ are respectively the Möbius and Riemann functions, cannot be analytically continued to the left half-plane, $\Re\,s\leq 0$ (in particular, we cannot give a reasonable evaluation of $P(-1)$), due to the clustering of poles along the imaginary axis arising from the nontrivial zeros of the Riemann $\zeta$ function.
Note the nasty-looking left edges in both plots above.
This result is originally due to Landau and Walfisz. See the linked papers for more details.