# Can someone please explain the Riemann Hypothesis to me… in English?

I’ve read so much about it but none of it makes a lot of sense. Also, what’s so unsolvable about it?

The prime number theorem states that the number of primes less than or equal to
$x$ is approximately equal to $\int_2^x \dfrac{dt}{\log t}.$ The Riemann hypothesis gives a precise answer to how good this approximation is; namely, it states that the difference between the exact number of primes below $x$, and the given integral, is (essentially) $\sqrt{x} \log x$.

(Here “essentially” means that one should actually take the absolute value of the difference, and also that one might have to multiply $\sqrt{x} \log x$ by some positive constant. Also, I should note that the Riemann hypothesis is more usually stated in terms of the location of the zeroes
of the Riemann zeta function; the previous paragraph is giving an equivalent form, which may be easier to understand, and also may help to explain the interest of the statement. See the wikipedia entry for the formulation in terms of counting primes, as well as various other formlations.)

The difficulty of the problem is (it seems to me) as follows: there is no approach currently known to understanding the distribution of prime numbers well enough to establish the desired approximation, other than by studying
the Riemann zeta function and its zeroes. (The information about the primes
comes from information about the zeta function via a kind of Fourier transform.) On the other hand, the zeta function is not easy to understand; there is no straightforward formula for it that allows one to study its zeroes, and because of this any such study ends up being somewhat indirect.
So far, among the various possible such indirect approaches, no-one has found
one that is powerful enough to control all the zeroes.

A very naive comment, that nevertheless might give some flavour of the problem, is that there are an infinite number of zeroes that one must contend with, so there is no obvious finite computation that one can make to solve
the problem; ingenuity of some kind is necessarily required.

Finally, one can remark that the Riemann hypothesis, when phrased in terms of the location of the zeroes, is very simple (to state!) and very beautiful: it says that all the non-trivial zeros have real part $1/2$. This suggests that perhaps there is some secret symmetry underlying the Riemann zeta function that would “explain” the Riemann hypothesis. Mathematicians have had, and continue to have, various ideas about what this secret symmetery might be (in this they are inspired by an analogy with what is called “the function field case” and the
deep and beautiful theory of the Weil conjectures), but so far they haven’t managed to establish any underlying phenonemon which implies the Riemann hypothesis.