Relation between The Euler Totient, the counting prime formula and the prime generating Functions

Question

](https://i.stack.imgur.com/lKFS0.png) Relation between The Euler Totient, the counting prime formula and the prime generating Functions There is a formula for the ivisor sum hiih is one of the most useful propertes of the Euler Formula to fond a relations with prime counting Formula ans sum of primes. Proof : http://vixra.org/abs/1901.0046

Answer 1

You are not really linking $\varphi(n)$ and $\pi(n)$. You are just adding things that cancel out.

$$\pi(n)=\sum\limits_{p\mid n}p+\sum\limits_{p\leq n, p\,\nmid \,n}1+\sum\limits_{d\neq p\mid n}\varphi(d)-n$$ $$\pi(n)=\sum\limits_{p\mid n}p+\sum\limits_{p\leq n, p\,\nmid\, n}1+\sum\limits_{d\mid n}\varphi(d)-\sum\limits_{p\mid n}\varphi(p)-n$$ $$\pi(n)=\sum\limits_{p\mid n}p+\sum\limits_{p\leq n, p\,\nmid\, n}1+n-\sum\limits_{p\mid n}(p-1)-n$$ $$\pi(n)=\sum\limits_{p\mid n}p-\sum\limits_{p\mid n}p+\sum\limits_{p\leq n, p\,\nmid\, n}1+\sum\limits_{p\mid n}1+n-n$$ $$\pi(n)=\sum\limits_{p\leq n}1$$ $\pi(n)$ is hidden in the second term $$\sum\limits_{p\leq n, p\,\nmid\, n}1$$ You could do that with any formula. Was there any reason to split it that way?