# On Euler totient function sum

Let $q$ an arbitrary integer. Is there any chance of getting a bound like $$\underset{d\mid q}{\sum}\frac{1}{\phi\left(q/d\right)^{2}}\ll\frac{1}{\phi\left(q\right)^{2}}?$$

• By $a \ll b$, you mean "There is a $C$ such that $a \leqslant C\cdot b$"? – Daniel Fischer Jan 7 '14 at 16:27
• Doesn't $\phi(q)\to \infty$ as $q\to\infty$, while the LHS is at least $1$? – Pablo Rotondo Jan 7 '14 at 16:35
• How did your question arise? – Pablo Rotondo Jan 7 '14 at 16:50

Not sure if I understand the question, but if this helps at all: $$\sum_{d\mid n}\frac{1}{\phi(n/d)^2}=\prod_{p\mid n}(1+\frac{1}{\phi(p)^2}+\frac{1}{\phi(p^2)^2}+\frac{1}{\phi(p^3)^2}...+\frac{1}{\phi(p^{v_p(n)})^2})$$ $$=\prod_{p\mid n}(1+\frac{1}{(p-1)^2}(1+\frac{1}{p^2}+\frac{1}{p^4}+..\frac{1}{p^{2(v_p(n)-1)}}))$$ $$=\prod_{p\mid n}(1+\frac{1-p^{-2v_p(n)}}{(p-1)^2(1-p^{-2})})$$
$$\sum_{d\mid n}\frac{1}{\phi(n/d)^2}<\prod_{p}(1+\frac{1}{(p-1)^2(1-p^{-2})})<3.4$$