In the classic proof of Bertrand's postulate by Paul Erdős, he shows that $x\# < 4^x$ where $x\#$ is the primorial for $x$.
Is there any tighter upper bound for a given primorial $x\#$? Ideally, does anyone know if there are any relatively recent papers on estimating the upper bound of a primorial?
On
A commonly studied arithmetic function is $$ \vartheta(x):=\sum_{p\leq x}\log(p)=\log(x\#). $$
The Prime Number Theorem can equally well be stated in terms of $\pi(x)$, $\psi(x)$, and $\vartheta(x)$. The equivalence of the statements is not difficult to show.
The classical error term for the Prime Number Theorem implies that there are constants $C_1,C_2>0$ such that $$ x\#<e^x e^{C_1x\exp\left(-C_2\sqrt{\log(x)}\right)}. $$ Basically any improvement on the error term in PNT improves this upper bound.
For a fairly pleasant version, for $x>1,$ we have $$ \log(x \#) < x \left(1 + \frac{1}{2 \log x} \right). $$ This is Theorem 4, formula 3.15 in Rosser and Schoenfeld (1962), Illinois J. Math.
Then, for $x \geq 563,$ we have $$ \log(x \#) > x \left(1 - \frac{1}{2 \log x} \right). $$ This is Theorem 4, formula 3.14.
The title of the article is Approximate Formulas for some Functions of Prime Numbers. It is pages 64-94, volume 6. I think I just found it online at some point...
YES, LINK
I did a simple experiment with C++, each line is just the low R-S bound, then a prime p, then log(p#), then the high R-S bound.
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