Understanding a very elementary property of factorials

I've seen this stated in a few places.

If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} \psi\left(\frac{x}{m}\right).$$

It is used by Ramanujan here. It is used by Jitsuro Nagura here.

Can anyone provide a proof for why it is true or provide a link to a proof?

Thanks very much.

We find

$$\sum_{m\ge1}\psi\left(\frac{x}{m}\right)=\sum_{m\ge1}\sum_{k\ge1}\vartheta\left(\sqrt[k]{\frac{x}{m}}\right)=\sum_{m\ge1}\sum_{k\ge1}\sum_{p\le \sqrt[k]{x/m}}\log p$$

$$=\sum_{m\ge1}\sum_{k\ge1}\sum_{mp^k\le x}\log p=\log \prod_{p\le x}p^{\#\{(m,k):mp^k\le x\}}=\log x!$$

since when counting $\#\{(m,k):mp^k\le x\}$, one sees for every $1\le n\le x$ there are $t=v_p(n)$ different tuples $(m,t),(mp,t-1),\cdots,(mp^{t-1},1)$ counted in the set (note $k\ge1$).

• Could you please expand a bit on your last paragraph? It looks to me like you're counting only the pairs $(m,v_p(n))$ with $m = 1$. – A.P. Apr 28 '15 at 9:13
• @A.P. Sorry, it's been two years so I don't quite remember what went wrong, but I suspect my brain might have went dark when typing out what I saw in my head. I've edited to give the correct list of tuples corresponding to each $1\le n\le x$. Basically, compute the sizes of the fibers of the obvious map $\{(m,k):mp^k\le x\}\to\{n:1\le n\le x\}$ given by $(m,k)\mapsto mp^k$ then sum the fibers' sizes. – anon Apr 28 '15 at 10:56
• Thank you, now I understand. It still took me a while to parse your wording, so for the benefit of other here is how I see it: for every positive integer $n$ there are exactly $v_p(n)$ ways of writing $n$ as $m p^t$ with $m \in \Bbb{Z}$ and $t \geq 1$. – A.P. Apr 28 '15 at 11:06

A slightly different variation based upon the prime factorisation of $$x!$$.

We obtain for integers $$x>0$$ \begin{align*} \color{blue}{\log x!}&=\log\prod_{p\leq x}p^{\left\lfloor\frac{x}{p}\right\rfloor+\left\lfloor\frac{x}{p^2}\right\rfloor+\cdots}\tag{1}\\ &=\log\prod_{p\leq x}p^{\sum_{m=1}^\infty\left\lfloor\frac{x}{p^m}\right\rfloor}\tag{2}\\ &=\sum_{p\leq x}\sum_{m=1}^\infty\log p\left\lfloor\frac{x}{p^m}\right\rfloor\tag{3}\\ &=\sum_{m=1}^\infty\sum_{p\leq \sqrt[m]{x}}\log p\left\lfloor\frac{x}{p^m}\right\rfloor\tag{4}\\ &=\sum_{m=1}^\infty\sum_{p\leq \sqrt[m]{x}}\log p\sum_{j=1}^{x/p^m}1\tag{5}\\ &=\sum_{m=1}^\infty\sum_{j=1}^x\sum_{p\leq \sqrt[m]{x}}\log p\tag{6}\\ &\,\,\color{blue}{=\sum_{j=1}^x\psi\left(\frac{x}{j}\right)} \end{align*} and the claim follows.

Comment:

• In (1) we do the prime factorisation of $$x!$$ and observe that $$\left\lfloor\frac{x}{p}\right\rfloor$$ counts the numbers $$\leq x$$ which are a multiple of $$p$$. Since we also have to add $$1$$ for each number which is a multiple of $$p^2$$ we add $$\left\lfloor\frac{x}{p^2}\right\rfloor$$, etc.

• In (2) we use the series notation. Note the series is finite.

• In (3) we use properties of the logarithm.

• In (4) we exchange the order of the series.

• In (5) we write the factor $$\left\lfloor\frac{x}{p^m}\right\rfloor$$ as sum.

• In (6) we exchange the order of the series again.

• +1. It's a nice job. – Felix Marin Dec 11 '18 at 4:07
• @FelixMarin: Thanks Felix. This answer follows a proof which was given by E. Landau in 1909. – Markus Scheuer Dec 11 '18 at 5:11
• I didn't know that. Thanks because everyday we learn something new. – Felix Marin Dec 11 '18 at 16:22
• @FelixMarin: You're welcome. The same holds true for me. :-) – Markus Scheuer Dec 11 '18 at 16:29