Convergence of alternating series based on prime numbers

Question

I've been experimenting with some infinite series, and I've been looking at this one, $$\sum_{k=1}^\infty (-1)^{k+1} {1\over p_k}$$ where $p_k$ is the k-th prime. I've summed up the first 35 terms myself and got a value of about 0.27935, and this doesn't seem close to a relation of any 'special' constants, except maybe $\frac12\gamma $.

My question is, has the sum of this series been proven to have a particular closed form? If so, what is this value?

http://forums.xkcd.com/viewtopic.php?f=17&t=59644&view=previous — 2'5 9'2, Nov 21 '12 at 03:33
More directly, see "Prime sums" on MathWorld, or entry A078437 in the Sloane's OEIS. — Sasha, Nov 21 '12 at 03:35
Here are some first $8250$ digits. http://dl.dropbox.com/u/5805435/Alternating_prime_sum.txt Unfortunately, it is not $\dfrac{\gamma}2$ — , Nov 21 '12 at 03:43
The Inverse Symbolic Calculator http://isc.carma.newcastle.edu.au/index has no suggestions. — Gerry Myerson, Nov 21 '12 at 04:48

score 3 · Accepted Answer · answered Dec 03 '12 at 03:05

As mentioned, this series has an expansion given by the OEIS. This series is mentioned in many sources, such as Mathworld, Wells, Robinson & Potter and Weisstein.

These sources all seem to imply that, though the series converges, no known "closed form" for this sum exists.

Convergence of alternating series based on prime numbers

1 Answers1

Linked