The Seductive Li(x)

The Riemann Hypothesis is equivalent to the assertion that

pi(x) = Li(x) + O(sqrt(x) . log x)

where Li(x) is Gauss's integral of 1/log x, pi(x) is the prime counting function, and x is sufficiently large.

What the expression states is that at some point in the natural number sequence and beyond that point, Li(x) estimates the number of primes less than any given number reasonably close to square root accuracy.

To get a grip on just how accurate that is, consider taking the census of the Chicago Metropolitan area which is about 6.5 million. You would not expect to get within the accuracy required by sqrt(x) . log (x) which is about 40,000 people. 40,000 is an error less than 1%.

Another way to think about Li(x) is in terms of the elementary result that prime gaps can be any size whatsoever. That is you can choose any huge number n, and there will be consecutive primes p and p' such that there is a sequence of n composite numbers between p and p'.

That means that at some point when x is a very large number, Li(x) estimates pi(p) and pi(p') and all the composite numbers in between to within sqrt(x) log x accuracy. There is this long stretch of numbers before pi(x) jumps to its next value and Li(x) just happens to be a good estimate for all the big values of pi(x) before the jump.

That is how the Riemann Hypothesis has seduced me. Wondering if it is really possible for Li(x) to do that kind of thing, to be that good.