TITLE: The function $\sum_{0}^{\infty} x^n/n^n$
QUESTION [28 upvotes]: The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I know of no name for it, nor any use for it aside from pedagogical, so this is a pure curiosity question which I hope is acceptable.
The function seems to have one real zero around $x = -1.40376$; a single extremum, a minimum around $x = -5.71837$; and then to approach the $x$-axis from below asymptotically as $x$ goes to negative infinity. Is this true?

REPLY [11 votes]: There is a paper of G. H. Hardy, where this function is studied in great detail:
G. H. Hardy, On the integral function $ \Phi_{ a,\alpha,\beta}(z)=\sum x^n/(n+a)^{\alpha n+\beta}$, Quarterly J. Math., 5 (1906) 37, 369-378. (Collected papers of G. H.Hardy, vol. IV, p. 128).