# 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:

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 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.

• The limits in (1) and (2) are infinite by straightforward comparisons; I don't know what you mean by (3) since the series does not converge for any value of $s$ near 0. – Mike Jury Aug 27 '14 at 12:48
• I think (3) is the best bet for anything to work. Once a functional equation is established for the Dirichlet series I would imagine that routine-ish techniques would take over from there. The functional equation would be the tricky part though. – BSteinhurst Aug 27 '14 at 12:57
• @MaxMuller (off-topic) I happened to notice that your user page links to an earlier user page. If you need your accounts merged, that can be done by filling out the form here mathoverflow.net/contact – j.c. Aug 27 '14 at 13:14
• @j.c. Ah ok, thank you for the tip! – Max Muller Aug 27 '14 at 13:21
• There is a difference between Abel summation and analytic continuation of power series. Abel summation requires that the power series is already analytic in the unit disc. – GH from MO Aug 27 '14 at 14:37