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 <a href="http://en.wikipedia.org/wiki/Zeta_function_regularization">regularization</a>. (The result was originally due to Landau and Walfisz, see <a href="http://link.springer.com/article/10.1007%2FBF03014596">this</a> paper. Froberg later showed it as <a href="http://link.springer.com/article/10.1007%2FBF01933420">well</a>.)

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: 

 1. 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?
 2. Lindelöf summation: Does $\lim_{x \to 0} \sum_{n=1}^{\infty} p_{n} n^{-nx} $ exist?
 3. 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 <a href="http://en.wikipedia.org/wiki/Divergent_series">method</a> 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.