a simple asymptotic (see for example [@Polya4Asymp], or [@GoddardAsymptotic] for more terms): $$\sigma_k = \sqrt{\frac{6}{k \pi}} \left(1 + O(1/k)\right)$$ as $k \rightarrow \infty$.

We may interpret $\sigma_k$ combinatorially. If $f_k$ is the probability density function of a sum of $k$ independent random variables distributed uniformly on $[-1/2, 1/2]$, then $\sigma_k = f_k(0)$. Thus, the form of the asymptotic for $\sigma_k$ is not surprising, in view of the Central Limit theorem.

\[def:definition\_of\_phi\] For $t_1, \dots, t_k$ positive real numbers, define the function $\Phi = \Phi_k$ by $$\label{eqn:definition_of_phi}
\Phi(t_1, \dots, t_k) = (I_{t_1} \ast \dots \ast I_{t_k})(0).$$

In particular, $\sigma_k = \Phi_k(1, \dots, 1)$.

There is an explicit formula for $\Phi$. In general, we have $$\label{eqn:phi_closed_form_k_var}
\Phi_k(t_1, \dots, t_k) = \frac{1}{(k-1)!2^k}\sum_{{\varepsilon}\in \{\pm 1\}^{k}} \omega ({\varepsilon}) ({\varepsilon}\cdot \mathbf{t})^{k-1} \operatorname{sgn}({\varepsilon}\cdot \mathbf{t}),$$ where $\omega({\varepsilon}) = \prod_i {\varepsilon}_i$ and ${\varepsilon}\cdot \mathbf{t} = \sum_i {\varepsilon}_it_i$ and $\operatorname{sgn}$ denotes the sign function. This is established in [@BorweinIntegral].

For $k = 3$, we can write (for $t_1 {\leqslant}t_2 {\leqslant}t_3$) $$\label{eqn:phi_closed_form_3_var}
\Phi_3(t_1,t_2,t_3) = \begin{cases}
t_1t_2 & t_3{\geqslant}t_1+t_2\\
\dfrac{2(t_1t_2+t_2t_3+t_1t_3) - (t_1^2+t_2^2+t_3^2)}{4} & \text{ otherwise}.
\end{cases}$$

In analogy with Proposition \[prop:twelfth\], we have the following.

\[prop:twelfth\_k\_var\] Let $k {\geqslant}3$ be an integer. For any equation $a_1x_1 + \dots + a_kx_k = 0,$ 