Inspired by Me

Calculus Level 3

Let

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

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 9.

1 solution

Sudeep Salgia
Oct 29, 2015

$\displaystyle S = \prod_{p \in \mathbb{A}} \left( 1 - \frac{1}{p^2} \right)$ where, $\mathbb{A}$ is the set of prime numbers.

Also using the Euler Product Formula , we can write the Riemann Zeta Function as, $\displaystyle \zeta (s) = \prod_{p \in \mathbb{A}} \left( \frac{1}{1 - p^{-s}} \right) = \frac{1}{S}$ .

Thus, $\displaystyle S = \frac{1}{\zeta{(2)}} = \frac{6}{\pi^2}$ .

On comparing and evaluating, we get $\displaystyle A+B+C = \boxed{9}$

First time I saw your solution! Couldn't resist myself from upvoting :P. Nice solution.

Aditya Kumar - 5 years, 7 months ago

Thanks! I agree I have posted a small number of solutions but it is strange you encountering this as the first one.

Sudeep Salgia - 5 years, 7 months ago

Inspired by Me

The answer is 9.

1 solution

0 pending reports