# Two $\psi$ functions

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!

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From Hardy and Wright, fifth edition paperback, page 340, the beginning of Chapter 22 "The Series of Primes (3)" $$\psi(x) = \sum_{p^m \leq x} \; \log p$$ with example $$\psi(10) = 3 \log 2 + 2 \log 3 + \log 5 + \log 7.$$ Then they say that $$\psi(x) = \log U(x)$$ where $U(x)$ is the least common multiple of all numbers (positive integers) up to $x.$ Finally $$\psi(x) = \sum_{p \leq x} \; \left\lfloor \frac{\log x}{\log p} \right\rfloor \; \log p.$$ It is occasionally important to allow $x$ to be a real number, not necessarily an integer. The Prime Number Theorem is equivalent to $$\psi(x) \sim x,$$ or $$\lim_{x \rightarrow \infty} \; \frac{\psi(x)}{x} \; = \; 1,$$ which is Theorem 434 on page 362.