Zeta gcd

Number Theory Level 5

$\large \sum_{\gcd(n,2016)=1} \frac{1}{n^{4}} = \dfrac{a}{b}\pi^{c}$

The equation above holds true for positive integers $a,b$ and $c$ with $a,b$ coprime. Find $a+b+c$ .

Notation : $\gcd$ denotes the greatest common divisor function.

The answer is 196485.

1 solution

Otto Bretscher
Apr 19, 2016

Nice problem, Comrade! The proof of the Euler product formula offers a one-step solution:

$S=\left(1-\frac{1}{2^4}\right)\left(1-\frac{1}{3^4}\right)\left(1-\frac{1}{7^4}\right)\zeta(4)=\frac{2000\pi^4}{194481}$ . The answer is $\boxed{196485}$

Zeta gcd

The answer is 196485.

1 solution

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