# Closed form for a series involving the $\Gamma$ and $\zeta$ functions

I was just wondering wether one can derive a closed form for $$\sum_{n=1}^{\infty}\frac{1}{\Gamma\left(\frac{1}{n}\right)\zeta\left(1+\frac{1}{n}\right)}$$ Numerical simulation gives $$S=1.20154...$$

The product formula between $$\Gamma$$ and $$\zeta$$ gives $$\sum_{n=1}^{\infty}\frac{1}{n\cdot a_n} \text{ with } a_n=\int_{0}^{\infty}\frac{x^{\frac{1}{n}}}{e^x-1}\text{d}x$$ But I can't seem to go any further.

Any suggestion ? Is it even doable ?

• $S=1.20652522381311998892341483649465039450107654831603\ldots$ – metamorphy Dec 29 '18 at 9:52