ProofWiki:Sandbox

$ \newcommand{\Re}{\mathrm{Re}\,}  \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$We consider, for various values of $s$, the $n$-dimensional integral\begin{align}  \label{def:Wns}  W_n (s)  &:=   \int_{[0, 1]^n}     \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}\end{align}which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer\begin{align} \label{eq:W3k}  W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.\end{align}Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.