Riemann Hypothesis and the prime counting function

This article on the prime counting function mentions that the Riemann Hypothesis is equivalent to the statement $$|\pi(x)-\rm {li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657 $$ My question is, is this an if and only if statement (ie if one were proven would the other follow)?

Also, how was this number $2657 $arrived at?

Solution 1:

Any bound $|\pi(x)-\rm {li}(x)|\le f(x)$ with $f(x)=O(x^{1/2 + \epsilon})$ implies the Riemann hypothesis, and RH is equivalent to the existence of a bound of that type. The point of this particular $f(x)$ is that it is known to be provable from the Riemann hypothesis, and the parameters in the proof have been worked out explicitly (and might be some of the best ones currently available), so that the end result is not "$f(x) = A\sqrt{x}(\log x)^B$ for $x > C$ for some $A,B$ and $C$", but $A = \frac{1}{8 \pi}$, $B=1$, $C = 2657$. The $O()$ notation signifies that the limit of $\frac{\log f(x)}{\log x}$ is $1/2$; if the limit had been $\rho \in [\frac{1}{2},1]$ the (non-trivial) zeros of the Riemann zeta would have real parts in the interval $[1 - \rho, \rho]$. The same bound with any other values of $A,B,C$ would also have been equivalent to RH.

$C$ can be eliminated by inflating $A$ to cover the first $2656$ cases and then one would get a bound valid for all $x \geq 1$. This would be less informative, because the important thing is to minimize the exponent $B$ of the logarithm (which is related to the vertical distribution of the zeros), then the constant $A$ (which measures some finer aspect of the zero distribution, but I don't know if it has been articulated what that is). The cutoff $C$ is much less significant, because it is not an asymptotic, large-$x$, quantity, and can be affected by moving a single zero along the critical line.

Solution 2:

The proof by L. Schoenfeld shows where these numbers $x_0$ form the statements with $x\ge x_0$ come from. We have $x\ge 2657$ in $6.18$ of Corollary $1$ on page $339$. This comes from $$ |\theta(x)-x|\le \frac{1}{8\pi}\sqrt{x}\log(x)^2, x\ge x_0=599, $$ and $x_0=559$ comes form the proof in $(6.3)$. To be honest, it is not easy to keep track of the constants $x_0$, which you will see reading this paper. The idea is, that the statement is true for all $x\ge x_0$, and an explicit computation (if you are lucky) can show which explicit value for $x_0$ is possible.