# Ordinary problem

Number Theory Level 5

$S = \displaystyle \sum_{n=1}^{\infty} \dfrac{\sigma_{4}(n)}{n^{8}}$

Find $$\dfrac{\pi^{12}}{S}$$.

Notation: $$\sigma_x(n)$$ denotes the sum of the $$x^\text{th}$$ powers of the positive divisors of $$n$$.

Hint: Use Bernoulli numbers to calculate the value of Riemann Zeta function. .

×