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Calculus Level 4

\[{\chi \left( y \right) =\begin{cases} +1\quad \text{ if}\quad y\equiv 1 \pmod 4 \\ -1\quad \text{if}\quad y\equiv 3 \pmod 4 \end{cases}}\]

Define \(\chi \left( y \right) \) as shown above. If the value of

\[\large{\prod _{ \text{p is prime}, p\ge 3 }^{ }{ \frac { { p }^{ 2 } }{ { p }^{ 2 }-{ \left( \chi \left( p \right) \right) }^{ 2 } } } }\]

can be expressed as

\[\large{\frac { A{ \pi }^{ B } }{ C } }\]

for some positive integers \(A,B,C\) such that \(A\) and \(C\) are coprime. Find the value of \( (A+B) \times C\).

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