Inspired by Me - 2

Calculus Level 5

Let

$S = \left(1- \dfrac{1}{2^4} \right) \left(1- \dfrac{1}{3^4} \right) \left(1- \dfrac{1}{5^4} \right) \left(1- \dfrac{1}{7^4} \right) \ldots$

If the value of the above product $S$ can be represented as $\dfrac{A}{C \pi ^B}$ , where $A$ , $B$ and $C$ are positive integers and $A$ , $C$ are coprime to each other, find the value of $A+B+C$ .

Note: The product is taken over all primes, starting from $2,3,5,7,\ldots$ .

You may want to look up the Euler product formula to help you with this problem.

The answer is 95.

1 solution

Tanishq Varshney
Oct 29, 2015

It's damn easy if u know Euler product of prime.

$\displaystyle \prod_{p} (1-p^{-s})=\frac{1}{\zeta (s)}$

where $p$ is a prime number and $\zeta (s)$ is zeta function.

Here $s=4$ so $\large{\frac{1}{\zeta (4)}=\boxed{\frac{90}{\pi^4}}}$

Same method! @Surya Prakash could u edit the problem a bit. I just got to know that it was zeta function in a glimpse.

Aditya Kumar - 5 years, 7 months ago

Inspired by Me - 2

The answer is 95.

1 solution

0 pending reports