Proof of Ramanujan's identity

Question

I'm having trouble understanding Ramanujan's formula from his proof of Bertrand's postulate, namely: $$ \ln \lfloor x\rfloor!=\sum_{k=1}^{\infty}\psi\left(\frac{x}{k}\right) $$ where $ \ln x = \log_ex$. Could someone explain me step by step, how to prove the formula? Thank you in advance.

Answer 1

For this proof we will use the definition

$$\psi(x) = \sum_{n \le x} \Lambda(n)$$

and the identity

$$\log n = \sum_{d|n} \Lambda(n)$$

which is proven here.

Now

$$\log [x]! = \sum_{n \le x} \log n = \sum_{n \le x} \sum_{d|n} \Lambda(d)$$

$$ = \sum_{ed=n} \Lambda(d) = \sum_{e \le x} \sum_{d \le x/e} \Lambda(d)$$

$$ = \sum_{e \le x} \psi(x/e).$$