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

TL;DR

This paper explores the asymptotic behavior, complex plane properties, and error function relations of a generalized sum related to Sophomore's dream, providing new insights into its oscillations, zeros, and approximations.

Contribution

It extends the Sophomore's dream sum to a broader function, deriving asymptotics, complex plane behavior, and advanced approximations involving the error function.

Findings

Asymptotic exponential and inverse-logarithmic behaviors identified.

Oscillating behavior along the imaginary axis with specific period.

Non-trivial zeros located in the left complex half-plane.

Abstract

Sophomore's dream sum $S=\sum_{n=1}^\infty n^{-n}$ 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 $|t|$ 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$t_n \simeq 2(n-1/8)\pi e/a$ for $a \geq 1$. The results obtained describe analytical integration of the function $x^{tx}$.