## Divergence of imaginary part for power series

I need some help with a divergence problem.

Say I have an equation:

$sigma[1/n^s]$ for a complex "s" with a real part 0<Re(s)<1.

I know the series diverges, which is simple enough, but what I need help with is understanding whether the real part diverges to a more dense infinity than the imaginary part.

Basically can anyone prove that:

sigma[ $cos(bln(n))/n^a$]>sigma[ $sin(bln(n))/n^a$]

for limits with n=1 to n=infinity