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What is the value of the series?:

$$\sum_{k=1}^\infty\frac {\zeta(-k)} k$$

Where $\zeta(z)$ is the Riemann Zeta function and for every negative integer $n$ we have $\zeta(n)=-\frac {B_{n+1}} {n+1}$.

I know the identity:

$$\sum_{k=2}^\infty(-1)^k\frac {\zeta(k)} k=\gamma$$

(where $\gamma$ is the Euler-Mascheroni constant) which is pretty similar so I was thinking if the series I wrote converges too ? Is there any closed form for the result ?

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  • $\begingroup$ Given the specific relation between Bernoulli numbers and the Riemann $\zeta$ function at negative integers, the series diverges. $\endgroup$ – Lucian Apr 28 '15 at 15:48
  • $\begingroup$ @Lucian is there a method to make it converge ? Maybe with Ramanujan summation... $\endgroup$ – Renato Faraone Apr 28 '15 at 16:19

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