# Math Help - More on Sum of the divisors

1. ## More on Sum of the divisors

Let σ2(n) = ∑d|nd2. Show that ∑ σ2(n)/n4 = π 6/540

2. Originally Posted by mndi1105

Let $\sigma_2(n) = \sum_{d \mid n}d^2.$ Show that $\sum_{n=1}^{\infty} \frac{\sigma_2(n)}{n^4} = \frac{\pi^6}{540}.$
$\sum_{n=1}^{\infty} \frac{\sigma_2(n)}{n^4}=\sum_{n=1}^{\infty}\frac{1 }{n^4} \sum_{d \mid n} d^2=\sum_{d=1}^{\infty}d^2 \sum_{n=1}^{\infty} \frac{1}{(dn)^4}=\sum_{d=1}^{\infty} \frac{1}{d^2} \sum_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^2}{6} \times \frac{\pi^4}{90}=\frac{\pi^6}{540}. \ \ \Box$