It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was originally due to Landau and Walfisz, see this paper. Froberg later showed it as well.)

However, there are loads of other summation methods. I am wondering whether any of the following summation methods can sum the divergent series of primes. For example:

- Abel summation/analytic continuation of power series (what is the difference?): Does $\lim_{x \to 1^{-} } \sum_{n=1}^{\infty} p_{n} x^{n} $ exist?
- Lindelöf summation: Does $\lim_{x \to 0} \sum_{n=1}^{\infty} p_{n} n^{-nx} $ exist?
- Analytic continuation of Dirichlet series: Does $\lim_{s \to 0} \sum_{n=1}^{\infty} \frac{p_{n}}{n^{s}} $ exist?

Do any of these methods or another summation method for assigning a number to the sum of primes work? If so, please also indicate what the closed form of the corresponding function (for which the limit exists) is.