(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}} $

enter image description here Is my conjecture already proven?

| cite | improve this question | | | | |

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.

| cite | improve this answer | | | | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.