# Approximateing the number of n-free numbers using Riemann zeta zeros

i have a question , in terence tao's blog :

https://terrytao.wordpress.com/…/continuous-approximations…/

he introduced a really nice way to approximate the partial sums of arithmetic functions , and his way was similar but easier than Riemann's way which is harder because it uses complex analysis but gives a little bet better results , but the two ways have main shared property which is the use of zeta function's zeros and dirichlet series , i'm writing a formula for approximating the number of n-free numbers before x :$F_n(x)$ in terms of zeta zeros , the dirichlet series of such a function is $\frac{\zeta(s)}{\zeta(ns)}$ and the problem that face me is , what are the residues of poles of $\frac{\zeta(s)}{\zeta(ns)}$ and how to find them

• $\zeta(s)$ has a Weierstrass product! – Jack D'Aurizio Feb 1 '18 at 12:59
• yeah i know , but i don't know how to generally find the residues of poles of any complex-vauled function $f(z)$ – Z. Dencker Feb 1 '18 at 14:13