The Riemann Hypothesis - one of the outstanding unsolved problems in math

The Riemann Hypothesis - one of the outstanding unsolved problems in math
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I wouldn't read into it any more than that.

The primes act very much like they were a random set of numbers with n having probability 1/log(n) of being prime. (Mathematicians always mean "natural log" when they say log.) Except, of course, that this is obviously false. Because, for example, there is only one even prime.

It is more accurate to say that the primes act very much like they were {2} plus a random set of odd numbers with n having probability 2/log(n) of being prime if it is odd. Except, of course, that this is obviously false. Because, for example, there is only one prime that is divisible by 3.

One can continue refining this statement in the obvious way of special casing more and more primes. Other than the obvious corrections that come from these refinements, anything that you'd conjecture from this description is either known to be true, or believed to be true. The number of primes below n. The number of primes below n in an arithmetic sequence. The density of twin primes. And so on, and so forth.

Now let's focus on the average of primes below n. (Mathematicians call that number pi(n).) If the primes all were independently random, then you'd expect that to be close to the sum, for i in 2 to n, of 1/log(i), which is approximately the integral from 2 to n of 1/log(x). The usual name for that integral is li(x). li(x) is approximately n/log(n). This estimate is known to be true. That's called the prime number theorem.

One of many equivalent forms of the Riemann Hypothesis is that pi(n) - li(n) is O(sqrt(n)). From the "random prime" hypothesis we can see why this is reasonable. It turns out that for large n, (pi(n) - li(n))/sqrt(li(n)) should follow a standard normal distribution. From which it is easy to show that the Riemann Hypothesis should have 100% odds of being true.

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