Riemann Zeta Function of 4/Proof 5

Proof
Zeta of 4 can be seen along the main diagonal in the illustration below.

The upper right and lower left quadrants are equal and are each equal to the sum associated with the 4th power term in the sin(x)/x expansion Basel_Problem/Proof_2


 * $\begin{array}{r|cccccccccc}

\displaystyle \paren {\map \zeta 2}^2 & \paren {\dfrac {1} {1}} & \paren {\dfrac {1} {4}} & \paren {\dfrac {1} {9}} & \paren {\dfrac {1} {16}} & \paren {\dfrac {1} {25}} & \paren {\dfrac {1} {36}} & \cdots \\ \hline

\paren {\dfrac {1} {1}} & \paren {\dfrac {1} {1}}^2 & \paren {\dfrac {1} {1}} \paren {\dfrac {1} {4}} & \paren {\dfrac {1} {1}} \paren {\dfrac {1} {9}} & \paren {\dfrac {1} {1}} \paren {\dfrac {1} {16}} & \paren {\dfrac {1} {1}} \paren {\dfrac {1} {25}} & \paren {\dfrac {1} {1}} \paren {\dfrac {1} {36}} & \cdots \\

\paren {\dfrac {1} {4}} & \paren {\dfrac {1} {4}} \paren {\dfrac {1} {1}} & \paren {\dfrac {1} {4}}^2 & \paren {\dfrac {1} {4}} \paren {\dfrac {1} {9}} & \paren {\dfrac {1} {4}} \paren {\dfrac {1} {16}} & \paren {\dfrac {1} {4}} \paren {\dfrac {1} {25}} & \paren {\dfrac {1} {4}} \paren {\dfrac {1} {36}} & \cdots \\

\paren {\dfrac {1} {9}} & \paren {\dfrac {1} {9}} \paren {\dfrac {1} {1}} & \paren {\dfrac {1} {9}} \paren {\dfrac {1} {4}} & \paren {\dfrac {1} {9}}^2 & \paren {\dfrac {1} {9}} \paren {\dfrac {1} {16}} & \paren {\dfrac {1} {9}} \paren {\dfrac {1} {25}} & \paren {\dfrac {1} {9}} \paren {\dfrac {1} {36}} & \cdots \\

\paren {\dfrac {1} {16}} & \paren {\dfrac {1} {16}} \paren {\dfrac {1} {1}} & \paren {\dfrac {1} {16}} \paren {\dfrac {1} {4}} & \paren {\dfrac {1} {16}} \paren {\dfrac {1} {9}} & \paren {\dfrac {1} {16}}^2 & \paren {\dfrac {1} {16}} \paren {\dfrac {1} {25}} & \paren {\dfrac {1} {16}} \paren {\dfrac {1} {36}} & \cdots \\

\paren {\dfrac {1} {25}} & \paren {\dfrac {1} {25}} \paren {\dfrac {1} {1}} & \paren {\dfrac {1} {25}} \paren {\dfrac {1} {4}} & \paren {\dfrac {1} {25}} \paren {\dfrac {1} {9}} & \paren {\dfrac {1} {25}} \paren {\dfrac {1} {16}} & \paren {\dfrac {1} {25}}^2 & \paren {\dfrac {1} {25}} \paren {\dfrac {1} {36}} & \cdots \\

\paren {\dfrac {1} {36}} & \paren {\dfrac {1} {36}} \paren {\dfrac {1} {1}} & \paren {\dfrac {1} {36}} \paren {\dfrac {1} {4}} & \paren {\dfrac {1} {36}} \paren {\dfrac {1} {9}} & \paren {\dfrac {1} {36}} \paren {\dfrac {1} {16}} & \paren {\dfrac {1} {36}} \paren {\dfrac {1} {25}} & \paren {\dfrac {1} {36}}^2 & \cdots \\

\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\

\end{array}$