I am looking for any information related to these functions: $$f(s)=\prod_{k=1}^\infty (1+\frac{1}{k^s})^k$$ $$g(s)=\prod_{k=1}^\infty k^{\frac{1}{k^s}}$$
Which converge for $\Re(s)>1$. I know that $\log(g(s))$ is a Dirichlet series with character $\chi(k)=\log(k)$. Besides that I don't really know where to look.