Sophomore's dream function: asymptotics, complex plane behavior and relation to the error function

Sophomore’s dream sum $S={\sum }_{n=1}^{\mathrm{\infty }}{n}^{-n}$ is extended to the function $f(t,a)=t{\int }_0^1(ax{)}^{-tx}dx$ with $f(1,1)=S$. Asymptotic behavior for a large $\mathrm{\mid }t\mathrm{\mid }$ is obtained, which is exponential for $t>0$ and $t<0,a>1$, and inverse-logarithmic for $t<0,a<1$. An advanced approximation includes a half-derivative of the exponent and is expressed in terms of the error function. This approach provides excellent interpolation description in the complex plane. The function $f(t,a)$ demonstrates for $a>1$ oscillating behavior along the imaginary axis with slowly increasing amplitude and the period of $2\pi iea$, modulation by high-frequency oscillations being present. Also, $f(t,a)$ has non-trivial zeros in the left complex half-plane with Im$t_n\simeq 2(n-1\mathrm{/}8)\pi e\mathrm{/}a$ for $a\ge 1$. The results obtained describe analytical integration of the function $x^{tx}$.