This is either a notation/history question or a point of confusion.
In (for example) Ramanujan's proof of Bertrand's postulate, he uses the following notation:
$\log [x]!$ means $\log ([x]!),$ in which $[x]$ is the floor function. This is clear because in one equation he has (omitting some stuff)
$\log\Gamma(x) \leq \log [x]! \leq \log \Gamma(x+1). $
He defines $\psi(x)$ as $\sum_{m=1}^\infty \vartheta(\sqrt[m]{x})$ in which $\vartheta(x) = \sum_{p \leq x} \log p . $ He claims (and let's assume) that $ \log [x]! = \sum_{m=1}^{\infty} \psi(x/m).$
Now the derivative of the function $\log\Gamma(x)$, in some places called the digamma function, is in most places denoted $\psi(x).$ For large x we have from the Wiki entry on "digamma" that $\psi(x) = \log x + O(1/x). $
So a check that this not (?) the same as the $\psi(x)$ above is that for the one above, by the PNT, we have $\psi(x)\sim x. $
Is this correct and if so can anyone explain how or why this happened? Or if it's wrong, point out the place where my confusion starts?
Thank you!