Two functions analogous to the factorial 
The factorial (red in the figure) is of course
$$n!=\prod_{i\in\mathbb{N},i\leq n} i$$
We could consider two analogous functions: the product of all primes below a given number (orange in the figure), and the least common multiple of all numbers below a given number (green in the figure). I came across these functions in a (failed) attempt to construct a counterexample for a certain conjecture.
My question is twofold


*

*Do these functions have names?

*Do there exist approximations for them, like Stirling's approximation for the factorial?

 A: The product of all primes not exceeding some number $n$ is called the primorial and denoted by $n\#$. Often that notation is extended to all positive real numbers, so
$$x\# := \prod_{p \leqslant x} p.$$
I am not aware of a special name for the least common multiple of all positive integers not exceeding $x$.
However, the (natural) logarithms of these functions are quite famous, they are the Chebyshev functions. The first Chebyshev function is given by
$$\vartheta(x) = \sum_{p \leqslant x} \log p = \log x\#$$
and the second Chebyshev function by
$$\psi(x) = \sum_{n \leqslant x} \Lambda(n) = \sum_{p\leqslant x} \biggl\lfloor \frac{\log x}{\log p}\biggr\rfloor\log p = \log \bigl(\operatorname{lcm}\,\{ n \in \mathbb{N} : 1 \leqslant n \leqslant x\}\bigr)$$
with the von Mangoldt function $\Lambda$, given by
$$\Lambda(n) = \begin{cases} \log p &\text{if } n = p^k\text{ with } k \geqslant 1 \text{ and $p$ prime,} \\ \quad 0 &\text{if $n$ is not a prime power}. \end{cases}$$
The Chebyshev functions are much used in prime number theory, their behaviour is more amenable to investigation using analytic methods than the behaviour of the prime counting function $\pi(x)$. But they have a close relation to the prime counting function, and results for one of these functions carry over to corresponding results for the others, e.g. via summation by parts.
The prime number theorem without error bounds, $\pi(x) \sim \frac{x}{\log x}$, is elementarily equivalent to each of $\vartheta(x) \sim x$ and $\psi(x) \sim x$. Error bounds for $\pi(x) - \operatorname{Li}(x)$ - where $\operatorname{Li}$ is the offset logarithmic integral, $\operatorname{Li}(x) = \int_2^x \frac{dt}{\log t}$ - correspond to similar error bounds for $\vartheta(x) - x$ and $\psi(x) - x$.
We do not know much about these differences so far. The currently best proven bound is
$$\pi(x) - \operatorname{Li}(x) \in O\Biggl(x\cdot \exp \biggl(-\frac{A(\log x)^{3/5}}{(\log \log x)^{1/5}}\biggr)\Biggr) \tag{$\ast$}$$
for some positive constant $A$. This implies the same type (one may need to use a different $A$) of bound for $\vartheta(x) - x$ and $\psi(x) - x$.
The error bound in $(\ast)$ is rather weak, it grows faster than $x^{1-\varepsilon}$ for every $\varepsilon > 0$. But it grows slower than $\frac{x}{(\log x)^k}$ for every $k\in \mathbb{N}$.
If the Riemann hypothesis is true, we have $\psi(x) - \operatorname{Li}(x) \in O(\sqrt{x}\,\log x)$, $\psi(x) - x \in O\bigl(\sqrt{x}\,(\log x)^2\bigr)$, and $\vartheta(x) - x \in O\bigl(\sqrt{x}\,(\log x)^2\bigr)$. Conversely each of these bounds would imply the Riemann hypothesis. In the other direction, Hardy and Littlewood proved $\psi(x) - x \notin o(\sqrt{x}\, \log \log \log x)$. Since $\psi(x) - \vartheta(x) \in O(\sqrt{x})$, the same holds for $\vartheta(x) - x$.
So we have approximations to $\vartheta(x)$ and $\psi(x)$, and consequently to the primorial and the least common multiple somewhat analogous to Stirling's approximation, but the bounds we have for these are much much weaker than the bounds in Stirling's approximation. Even if the Riemann hypothesis is proved some day, the bounds will remain much weaker.
