An elementary explanation of the Riemann Hypothesis

Elementary here is relative, but this explanation of what the Riemann Hypothesis means should be understandable to people who have no idea about complex analysis, zeta functions, and other usual things that you see in explanations of the Riemann Hypothesis. However I will invoke a substantial amount of probability theory.

There is a famous hypothesis in number theory. It is the basis of something known as Probabilistic Number Theory. It is trivially false. And yet so tantalizingly accurate. Here is the first form.

Every integer n has independent probability 1/log(n) of being prime. (Note that log(n) here means the natural log of n. Mathematicians always use that form, and never the log base 10.)

This is obviously false. We can improve it. The primes are 2, and then every odd integer n has independent probability 2/log(n) of being prime.

This is obviously false. We can improve it again. The primes are 2 and 3, and every number n not divisible by 3 with independent random probability (2/1)*(3/2)*(1/log(n)) of being prime.

Obviously we can improve this set of "random" conjectures infinitely often. And eventually will arrive at the right answer.

Now this is obviously false. Everyone knows that the primes are not, actually, random. Why did I say that it was tantalizingly accurate? Well because of the following.

All evidence available to us suggests that every statement that you would guess from the above sequence of conjectures is, in fact, true.

Let us take a simple example. Suppose that every integer n has independent probability 1/log(n) of being prime. Then on average how many primes would you expect to find up to N? It would be 1/log(2) + 1/log(3) + ... + 1/log(N). Let's call that function li(N). Let's call the count of primes up to N pi(n). (That is what number theorists call it. I wouldn't have called it that, but I'll follow tradition.) Here are some statements that are known to be true about pi(n) and li(n).

First of all li(n) is the integral, from 2 to N, of 1/log(x) plus O(1). That integral is n/log(n) + o(n/log(n)). You'd expect pi(n) to follow similar behavior. And, in fact, the prime number theorem says, pi(n) = n/log(n) + o(n/log(n)).

But we can do better than that. We expect pi(n) - li(n) to be some kind of weird random walk, centered around 0. If you know about random walks, you'd expect that expression to switch signs, infinitely often. Surprisingly, Littlewood proved this true in 1914. But the switches should be very, very rare. And, in fact, the first switch is believed to be around 1.39822×10³¹⁶.

Can we go on? What is the density of twin primes? Naively you might expect it to be (1/log(n))². But the version with 3 and 2 struck out tells you that in every 6 primes you have 2 numbers with probability (3/log(n))² of being prime. You can iteratively improve this estimate with 5, 7, 11, etc. This converges to a special case of the Hardy–Littlewood conjecture which would prove the twin prime conjecture. All existing numerical evidence points to this being true.

So from this probabilistic hypothesis we can come up with many, many statements. Some of which are known to be true. Some of which are not. But I have not yet connected this to the Riemann Hypothesis. Let me correct that.

Both the conjecture and Littlewood's theorem suggests that pi(N) - li(N) behaves a lot like a random walk. If that is so, what can we say about the errors? Well from the central limit theorem, we should expect pi(N) - li(N) to have something like a normal distribution. In fact there are many forms of the central limit theorem. The right one to invoke is the Lyapunov condition, from which pi(N) - li(N) can be shown to have average 0, and variance equal to the sum of the variances for each number being prime. If you look at the formula for a biased coin flip, the variance for n being prime turns out to be 1/log(n)-1/log(n)². If you sum up the variances up to N, the sum of the 1/log(n) terms dominates, and the variance turns out to be li(N) + o(li(N)) = N/log(N) + o(N/log(N)). From which the expected error is O(sqrt(variance)) = O(sqrt(N/log(N)).

Now the expected error may be that, but the error will occasionally take very unexpected values. However the law of the iterated logarithm says that the maximum deviation should be O(sqrt(N/log(N)) * log(log(sqrt(N/log(N))))) < O(sqrt(N)).

Now let me get in to the Riemann Hypothesis proper.

Usually when you see the Riemann Hypothesis it usually talks about nontrivial zeros of the Riemann Zeta function. Thanks to a bunch of complex analysis that I won't get in to, it turns out that all of the non-trivial zeros of the Riemann zeta function have real part < p if and only if pi(N) - li(N) = o(N^p). Well from the above analysis one would expect that for every p > 1/2, that there would be no non-trivial zeros of the Riemann zeta functions which are p or greater. It is also known that if non-trivial zeros are under 1/2, then some must be over 1/2, and therefore the above analysis (plus these facts I am giving you from complex analysis that I have not described) say that all of the non-trivial zeros should have real part exactly 1/2. Conversely if any non-trivial zero has real part greater than 1/2, then pi(N)-li(N) cannot obey the rules one would expect from probabilistic number theory.

Currently all existing evidence suggests that the Riemann hypothesis is true. It is known that the first 250 * 10⁹ roots of the Riemann zeta function have real part 1/2. It it known that infinitely many roots have real part 1/2. All existing numerical evidence we have suggests that the result is true. All that we are missing at this point is a pesky detail called a proof. :-)