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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,

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If you can calculate a few decimals, there are places to look things up. –  Gerry Myerson Aug 18 '11 at 1:24
    
None of (122)-(131) involve $\zeta'$, so it's not clear to me why those formulas lead you to think that sums with $\zeta'$ will be "nice". –  Gerry Myerson Aug 18 '11 at 3:56

1 Answer 1

up vote 8 down vote accepted

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 some, 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 ?

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+1, I like this answer. However the plouffe inverter is not very good at recognizing constants from number theory. For example, $B_3$ in equ (17) on this page:mathworld.wolfram.com/MertensConstant.html this constant does come up in a few places. –  Eric Naslund Aug 18 '11 at 20:56

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