Exploring Divisor Function 2

Number Theory Level 5

$\sum _{n=1}^{\infty}\frac{d\left(2015n\right)}{n^2}=\frac{A\pi^B}{C}$

Where $d(n)$ counts the number of divisors of $n$ . Find $A+B+C$

Hint: 2015 is a squarefree integer.

This problem is part of the set Exploring the divisor function

The answer is 177889577.

1 solution

Otto Bretscher
Feb 13, 2016

With $2015=5\times13\times31$ , we have $(2-\frac{1}{5^2})(2-\frac{1}{13^2})(2-\frac{1}{31^2})(\zeta(2))^2=\frac{31721473}{146168100}\pi^4$