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Calculus Level 5

Let $p$ be the set of prime numbers. If

$\prod _{ p }^{ }{ \left( \sum _{ n=0 }^{ \infty }{ { p }^{ -4n } } \right) } =\frac { { \pi }^{ A } }{ B } ,$ where $A, B$ are positive integers, find $A + B$ .

The answer is 94.

1 solution

Aditya Kumar
Aug 30, 2015

$Using\quad Euler's\quad product\quad rule,\quad we\quad get\\ \prod _{ n }^{ }{ \left( \sum _{ n=0 }^{ \infty }{ { p }^{ -4n } } \right) } =\quad \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } } =\zeta \left( 4 \right) =\frac { { \pi }^{ 4 } }{ 90 }$

@Aditya Kumar There are many edits which are required in the problem. Do check and edit as fast as you can (what are A and B?, and it should be $p$ under the product sign)

And, when it comes to proving Euler's Product rule, it's way easy as anyone can think but I don't think one should plagiarize and disrespect the great Euler(and his great proof). So, copying the same proof will not be a good thing to do, don't you think?

Kartik Sharma - 5 years, 9 months ago

I agree with Kartik Sharma. It's better prove that the zeta(4) = pi^4 /90 is also true. (Don't apply Bernoulli numbers!).

Pi Han Goh - 5 years, 9 months ago

That's the reason I didn't post the proof. I wanted to see if there were new ways to prove it or not. And sorry for the typo.

Aditya Kumar - 5 years, 9 months ago

This is beauty, not a beast!

The answer is 94.

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