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I want to compute the following sum over primes: $$\sum\limits_{p \text{ prime}}\sum\limits_{k=1}^\infty(\log(p^k))\left(\frac{1}{2p^k} - \Phi[-1,1,p^k]\right),$$ where $\Phi[z,s,a]$ is the Hurwitz-Lerch transcendent. Can I compute it using Mathematica? Here is the code I used

Sum[log((Prime[n])^k) ( Prime[n])^(-k)/2-HurwitzLerchPhi[-1,1,Prime[n]^k]), 
                                        {k, 1, Infinity}, {n, 1, Infinity}]
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migrated from scicomp.stackexchange.com Nov 9 '13 at 15:41

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Duplicate: Double sum over primes –  The Toad Nov 9 '13 at 15:58
You asked a similar question previously and ignored requests made to you to post the code you had developed. Again you post no code. Why should we help you? This is not a free coding service. –  m_goldberg Nov 9 '13 at 16:07
I suggest to leave open this question. Perhaps it might have a symbolic solution. Nonetheless the linked post provides possible ways for an approximate sum. –  Artes Nov 9 '13 at 21:50
@Artes I saw your comment just now... do you suggest reopening it? –  The Toad Nov 9 '13 at 22:11
@rm-rf I'd rather be more careful with this question. Probably this sum can be only approximated numerically, but I'm not sure. Perhaps someone might find an exact sum. I guess we would leave it open for one or two days. –  Artes Nov 9 '13 at 22:20

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