I have been researching equivalent statements on the Riemann Hypothesis. Schoenfeld proved that Riemann Hypothesis is equivalent to the bound $$|\pi(x) - {\rm Li}(x)| < \frac{1}{8\pi}\sqrt{x}\log x$$ for $x\geq 2657$. Excepting Schoenfeld's explicit bound, I have not been able to find explicit upper and lower bounds on the Prime counting function such that they would imply that Riemann Hypothesis is true.
In particular, I am wondering if they can exist $A$ and $B$ such that $$\frac {x}{\log(x)}+A<\pi(x)<\frac {x}{\log(x)}+B$$ and such that they would imply that Riemann Hypothesis is true.
Can these bounds exist? Are they known?
Thanks in advance!
I suspect you want $A$ and $B$ to be constants, since you don't indicate any dependence in them on $x$. So you appear to be asking if RH might be equivalent to $\pi(x) = x/\log x + O(1)$ with explicit constants in the role of the $O(1)$ bound.
If you look at the graph at the bottom of the page https://primes.utm.edu/howmany.html, where $\pi(x)$ is in red and $x/\log x$ is in green, what it suggests there is true: $\pi(x) - x/\log x \to \infty$ as $x \to \infty$. Specifically, the oldest form of the prime number theorem with error term $$ \pi(x) = {\rm Li}(x) + O(xe^{-c\sqrt{\log x}}) $$ implies $$ \pi(x) = {\rm Li}(x) + O_k\left(\frac{x}{(\log x)^k}\right) $$ for all $k > 0$, and integration by parts twice tells us $$ {\rm Li}(x) = \frac{x}{\log x} + \frac{x}{(\log x)^2} + O\left(\frac{x}{(\log x)^3}\right), $$ and together these estimates imply $$ \pi(x) = \frac{x}{\log x} + \frac{x}{(\log x)^2} + O\left(\frac{x}{(\log x)^3}\right), $$ so $\pi(x) - x/\log x \sim x/(\log x)^2$. Thus $\pi(x) - x/\log x \to \infty$ as $x \to \infty$.
In contrast to that, Littlewood proved that the difference $\pi(x) - {\rm Li}(x)$ changes sign infinitely often.
While the Riemann hypothesis is equivalent to $\pi(x) = {\rm Li}(x) + O(\sqrt{x}\log x)$, there is no estimate of the form $\pi(x) = x/\log x + O(x^{1-\delta})$ for some $\delta$ in $(0,1)$: using the last displayed estimate on $\pi(x)$ above, if $\pi(x) = x/\log x + O(x^{1-\delta})$ then $$ \frac{x}{(\log x)^2} = O\left(\frac{x}{(\log x)^3}\right) + O(x^{1-\delta}) = O\left(\frac{x}{(\log x)^3}\right) $$ for large $x$, which is false.
The reality is that the use of $x/\log x$ as an approximation for $\pi(x)$ works in the asymptotic estimate in the prime number theorem but not in non-asymptotic estimates that you will find in equivalent reformulations of the Riemann hypothesis. We usually write PNT in terms of $x/\log x$ because it has the attraction of being a simple expression in comparison to ${\rm Li}(x)$, but you have to give up on working with $x/\log x$ as an approximation to $\pi(x)$ when you want to look at more sophisticated things like RH.