Connection between the area of a n-sphere and the Riemann zeta function?

The relation $$\frac{\zeta(s)}{A(s)}=\frac{\zeta(1-s)}{A(1-s)}$$ stems from the fact that $$\frac{\zeta(s)}{A(s)}=\frac{1}{2}\mathcal{M}\left(\psi\right)\left(\frac{s}{2}\right),$$ where $\mathcal{M}$ denotes the Mellin transform and $$\psi(x)=\sum_{n=1}^\infty e^{-\pi n^2 x}.$$ This function is symmetric due to the functional equation for $\psi(x)$, and that yields the functional equation for the zeta function. The reason for the connection to the surface area appears in my previous answer https://math.stackexchange.com/a/1494471/6075, which provides a geometric explanation for integer dimensions - the case where we can actually interpret the surface area. In that answer, I showed that the factor of $A(s)$ appears due to the spherical symmetry of a naturally arising $s$-dimensional integral expression for $\zeta(s)$.

Remark 1: This gives an excellent way to remember the precise form of the functional equation!

Remark 2: Note that the surface area here, $A(s)$, corresponds to the $s-1$-dimensional sphere, that is the surface of the $s$-dimensional ball, whereas in the linked answer, $A_{k-1}$ is the surface area of the $k-1$-dimensional sphere, that is the surface of the $k$ dimensional ball.