Bounds of specific function involving primes So i was playing around with factorials and Primes lately and i came up with a (to me) new function, which is:
$$\prod_{p \leq x \text{, p is prime}} p^{\lfloor{log_p \lfloor x \rfloor \rfloor}}$$
Now this function is always less than the factorial and it's in some way or another exponential but i am having trouble finding an approximation or any kind of bounds to this function. So I would greatly appreciate some help.
 A: $$ n\leqslant\lfloor\log_p(x)\rfloor\iff n\leqslant \log_p(x)\iff p^n\leqslant x $$
Therefore $\lfloor\log_p(x)\rfloor=\max_{n\leqslant x}v_p(n)$ and, knowing that $\lfloor\log_p(x)\rfloor= 0$ for all $p>x$, we have
$$ \prod_{p\leqslant x}p^{\lfloor \log_p(x)\rfloor}=\prod_{p}p^{\lfloor \log_p(x)\rfloor}={\rm lcm}(1,\ldots,x) $$
A: Your final product $P$ is a product of powers of the different $p \leq x$ involved. Essentially, you've created the unique prime factorization of your final answer, with each exponent being $\lfloor \log_p x \rfloor$. Calling each exponent $n_p$, we can see that the final product will be:
$$P = 2^{n_2}3^{n_3}5^{n_5} \cdots$$
Now consider each term. $2^{n_2}$ is the greatest power of $2$ then is smaller than $x$. Same for $3^{n_3}$ and so on.
Since each separate multiplicand is a prime power, we can be sure that $P = \text{lcm} (2^{n_2}, 3^{n_3}, \cdots)$. But every integer $i \leq x$ must have a prime factorization in which each exponent is less than or equal to the set of $n$s. For instance, if $x = 26$, neither $26$ nor any smaller integer has a prime factorization with larger than $2^4$ in it, nor larger than $3^2, 5^2, 7^1 \cdots 23$.
But that list of prime powers is equal to $P$, as we saw earlier. And no composite number $4 \leq i \leq x$ has a larger prime power in its factorization, so $\text{lcm} (i, P) = P$. Therefore $P = \text{lcm}(2, 3, \cdots x)$, as others have said. Hopefully you'll find this explanation simpler or useful.
A: So after doing more research, I rediscovered the chebyshev function, which is equal to the natural logarithm of the function I was searching for and which gives decent upper and lower bounds for my search. Thank you for all the great answers, the lcm(1,...,x) is of course exactly what I was looking for initially.
