conjecture about Totient summatory function

Question

(I am sorry that I am poor at English)

Definition of Totient summatory function $\Phi \left ( n \right )$ is

$\Phi \left ( n \right )= \sum_{k=1}^{n}\varphi \left ( k \right )$

For a large number $n\geq 1$, I guess

$\frac{\Phi\left (n \right ) }{p_{n}} \sim \alpha n+\beta $

(${p_{n}}$ is n-th prime number)

I already calculated $\frac{\Phi\left (n \right ) }{p_{n}} $ (when $n\leq 1000$), Its graph is as follows $\frac{\Phi\left (n \right ) }{p_{n}} $

Is my conjecture already proven?

Answer 1

From $\varphi(n) = \sum_{d | n} \mu(d) \frac{n }{d}$ $$\sum_{n=1}^\infty \varphi(n) n^{-s} =\frac{\zeta(s-1)}{\zeta(s)}$$ Since $\frac{\zeta(s-1)}{\zeta(s)}$ has a dominating pole of order $1$ at $s=2$ and $\varphi(n) \ge 0$ we obtain $$\sum_{n < x} \varphi(n) \sim \frac{x^2}{\zeta(2)}$$ The PNT means $p_n \sim n \ln n$ so your asymptotic is not correct.