Now the proofs of the prime number theorem given by Hadamard and

de la Vallee Poussin actually show that not only is pi(x) asymptotically

equal Li(x), but moreover that pi(x) = Li(x) + an error term.

And they were able to give explicit balance on this error terms,

particularly on the growth rate of the error term.

So the error term does go to infinity as x goes to infinity, but

at a controlled rate, and the control was given by Hadamard and

de la Vallee Poussin in their proof.

Von Koch, in 1901, was able to give the best possible bounds on this error term.

Assuming the Riemman hypothesis is true, Schoenfeld made this precise and

proved that the Riemann hypothesis is equivalent to pi of x minus li of

x being bounded a buff by root of x times natural logarithm of x over eight pi.

Now li of x isn't quite the same as li of x up here, here is a lowercase l.

And that's not a typo.

Up here there's a uppercase L.

So lowercase li of x is the un-offset logarithmic integral function.

So it's the integral of zero to x of 1 over ln t dt.

And it's related to the offset logarithmic integral function,

the Uppercase Li of x is lowercase li of x minus lowercase li of 2.

So this result right here is only true if the Riemann hypothesis is true,

and moreover, it is equivalent to the Riemann hypothesis.

And it tells us a lot about the distribution of prime numbers.

The voracity of the Riemann hypothesis,

therefore implies results about the distribution of prime numbers,

in particular about how regularly they're distributed about their expected

locations and how much they cannot vary from their expected locations.