Is it possible to sum the divergent series with prime coefficients? 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. 
 A: This question is  poorly formulated, so I am not sure if it can be properly answered. I have some comments, rather then an answer. First, the methods 1,2 
(Abel and Lindelöf summation), as well as any other linear regular summation method with positive matrix, are useless when dealing with a series with positive summands: if the original series is not convergent, then the summation does not help. Note that almost all of the standard methods are of this kind.
Analytic continuation methods, including zeta function regularization, is another story. But this is fishy: even if the continuation exists, it may well depend on the path. Besides, which function to continuate? I am at the opinion that a question like "Do any of these methods work?", in the context of analytic continuation, simply does not make any sense. What makes sense is "Does THIS method work?", and the choice of the method should be well motivated.  Of course, motivation here may depend on the purpose of calculating the sum of a series, not only on the series itself: for example, zeta function regularization is a reasonable choice for a calculation of the Casimir force, because it can be shown to produce a correct physical result. 
