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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 $(A+B) \times C$ .

1 solution

Isaac Buckley
Jan 15, 2016

Notice that $\chi(p)^2=1$ for all prime $p$ so we can simply the above expression to:

$\Large \prod_{p\neq 2} \frac{1}{1-\frac{1}{p^2}}=\frac{3}{4}\zeta(2)=\frac{\pi^2}{8}$

It would be helpful to others (beginners) if you include the reference to Euler product which is the main thing used in the solution.

Prasun Biswas - 5 years, 5 months ago