Closed form for $\sum_{n=2}^\infty \frac{1}{n^2\log n}$

Question

I had attempted to evaluate

$$\int_2^\infty (\zeta(x)-1)\, dx \approx 0.605521788882.$$

Upon writing out the zeta function as a sum, I got

$$\int_2^\infty \left(\frac{1}{2^x}+\frac{1}{3^x}+\cdots\right)\, dx = \sum_{n=2}^\infty \frac{1}{n^2\log n}.$$

This sum is mentioned in the OEIS.

All my attempts to evaluate this sum have been fruitless. Does anyone know of a closed form, or perhaps, another interesting alternate form?

Answer 1

The closed form means an expression containing only elementary functions. For your case no such a form exists. For more informations read these links:

http://www.frm.utn.edu.ar/analisisdsys/MATERIAL/Funcion_Gamma.pdf

http://en.wikipedia.org/wiki/Hölder%27s_theorem

http://en.wikipedia.org/wiki/Gamma_function#19th-20th_centuries:_characterizing_the_gamma_function

http://divizio.perso.math.cnrs.fr/PREPRINTS/16-JourneeAnnuelleSMF/difftransc.pdf

http://www.tandfonline.com/doi/abs/10.1080/17476930903394788?journalCode=gcov20

Some background are needed for your understanding and good luck with these referrences.