I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants:
$$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$
and
$$\sum_{k=2}^{\infty}-\frac{\mu(k)}{k}\frac{\zeta^{'}(k)}{\zeta(k)}.$$
It is not hard to check that both converge absolutely, but based on (122)-(131) of the Wolfram Zeta Page, I think both sums should take on nice values.
References to any material which deals with these types of sums in general is also appreciated.
Thanks,
Not really an answer to the question posed, but too big to be a comment.
The value of the first sum is
$$ \sum_{k>=2} - \frac{\zeta^\prime(k)}{\zeta(k)} = 0.850312379764164578438788712404715501868902645375196564818394 $$
The above value is not recognized by Plouffe's inverter, unfortunately.
Moreover, because $\log \zeta(s) = - \sum_{k\ge1} \log (1-p_k^{-s})$ for $s>1$, it follows that $$ - \sum_{k>=2} \frac{\zeta^\prime(k)}{\zeta(k)} = \sum_{i \ge 1, k\ge 2} \frac{\log p_i}{p_i^k -1} $$ and even $\sum_{k>=1} (x^k-1)^{-1}$ is not known in closed form, so the chances are slim, but one never knows.
In regard to the other sum, it comes close to
$$ \zeta_P(s) = \sum_{k \ge 1} p_k^{-s} = \sum_{n\ge 1} \frac{\mu(n)}{n} \log \zeta( s n) \qquad \text{ for } s > 1 $$
Differentiating with respect to $s$ and subtracting the pole term we would get
$$ \lim_{s \to 1+} \zeta_P^\prime(s) - \frac{1}{1-s} = C + \sum_{k \ge 2} \mu(k) \frac{\zeta^\prime(k)}{\zeta(k)} $$
which, again, is not quite the same. Numerical value for the second sum also does not turn up any results in Plouffe's inverter:
$$ \sum_{k \ge 2} \frac{\mu(k)}{k} \frac{\zeta^\prime(k)}{\zeta(k)} =0.344146097673912783894171441679617569043972324522437879896534 $$
Why do you expect these sums to have nice values ?