I wonder why mathematics are like this, even you study and study you can't succeed if you don't have some insights to make the tricks. It's not like some other sciences, you read the information very well and you'll be good, you'll have a A+ on your exams, just reading, but not in math, if you don't come up with the ideas on your exam, you're lost xD

Define $\beta_s$:
$$\beta_s(x)=\sum_{n=1}^\infty\frac{1}{n^{s}+x^s}$$

Then if $xy=1$,

$$\sum_{m=1}^\infty\frac{\beta_s(mx)}{m^s}+\sum_{m=1}^\infty \frac{\beta_s(my)}{m^s}=\zeta(s)^2$$

For all real $xy=1$,
$$\frac{\pi^3}{180}(x^2+y^2+5)-\frac{1}{2}\zeta(3)(x+y)=$$

$$\frac{1}{x}\sum_{m=1}^\infty\frac{1}{m^3(e^{2\pi mx}-1)}+\frac{1}{y}\sum_{m=1}^\infty\frac{1}{m^3(e^{2\pi my}-1)}$$