# 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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## 1 Answer

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.

The digamma function is something else. There is nothing much to be done about this. There are not enough letters to go around, people doing rather different topics choose the same letters, if enough time passes the same letter gets established as standard for more than one concept. That's life.

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