Logarithm of factorial equal to sum of logarithm of primes

Question

Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that \begin{align} \log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\rfloor \cdot \log(p) \end{align}

To do it, I'd like to prove a more general statement, which says \begin{align} \sum_{n\leq x} \log(n) = \sum_{p^{\nu}\leq x \\ p\in \mathcal{P}} \left\lfloor\dfrac{x}{p^{\nu}}\right\rfloor \cdot \log(p) \qquad \forall\, x\geq1 \end{align}

$\textbf{My attempt}$:

I tried to using the Fundamental Theorem of Arithmetics to write every $n\in \mathbb{N}^{*}$ as product of primes, i.e. \begin{align} n= \prod_{p\in P \\ v\geq 0} p^{\nu} \ \end{align} Then, \begin{align} \log(n) = \sum_{p\in \mathcal{P}\\ v\geq 0} \nu \cdot \log(p) \qquad (\ast) \end{align} Now I have some problem to rewrite $\sum_{n\leq x} \log(n)$ using $(\ast)$. I tried to find an equivalent condition to $n\leq x$ whithout success.

Any suggestions? Thanks in advance!

Answer 1

Hint: Instead of taking each integer of $N!=\prod_{n=1}^N n$ separately, take them all together and consider each prime on the entire factorial. Specifically, show that the largest $E$ such that $p^E\mid N!$ is $$ \sum_{\nu=1}^\infty\left\lfloor\frac N{p^\nu}\right\rfloor $$

Answer 2

Note that $\lfloor \frac{N}{p^\nu} \rfloor$ counts the number of multiples of $p^\nu$ that are at most $n$. Note that every multiple of $p$ contributes a factor of $p$ in $N!$, giving $\lfloor \frac{N}{p} \rfloor$ in the exponent of $p$. But every multiple of $p^2$ contributes an additional factor not already accounted for, giving an additional $\lfloor \frac{N}{p^2} \rfloor$ in the exponent. Repeating, we find that the exponent of $p$ is $\sum \limits_{p^\nu \le n} \lfloor \frac{N}{p^\nu} \rfloor$. Decomposing $N!$ into its prime factorisation now finishes the problem.

For further information, see the following: https://en.m.wikipedia.org/wiki/Legendre%27s_formula