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?

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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.

A direct translation of RH (Riemann Hypothesis) would be very baffling in layman’s terms. But, there are many problems that are equivalent to RH and hence, defining them would be actually indirectly stating RH. Some of the equivalent forms of RH are much easier to understand than RH itself. I give what I think is the most easy equivalent form that I have encountered:

The Riemann hypothesis is equivalent
to the statement that an integer has
an equal probability of having an odd
number or an even number of distinct
prime factors. (Borwein page. 46)

You have to…have to….read this friendly introduction. (Making it CW since it’s just a link.)

In very layman’s terms it states that there is some order in the distribution of the primes (which seem to occur totally chaotic at first sight). Or to say it like Shakespeare: “Though this be madness, yet there is method in ‘t.”

If you want to know more there is a new trilogy about that topic where the first volume has just arrived:

It is a marvelous and easy to understand book from a number theorist who knows his stuff!

Here is a very simply description of the Riemann Hypothesis that requires nothing more than a 3rd grade education to understand:

There is also a beautiful proof linking the Farey sequence of fractions to the Riemann hypothesis by Jerome Franel. It’s only three pages long and should be able to be understood by any undergraduate mathematics major.

Let $H_n$ be the nth harmonic number, i.e. $ H_n = 1 + \frac12 + \frac13 + \dots + \frac1n.$ Then, the Riemann hypothesis is true if and only if

$$ \sum_{d | n}{d} \le H_n + \exp(H_n)\log(H_n)$$

It is straight forward. We have a Zeta function, ‘analytically continued’ that says

$\zeta(s)[1-2/2^s] = 1 – 1/2^s + 1/3^s – 1/4^s + ……$ Here s is a complex variable. Thus $s=\Re(\sigma) + \Im(\omega).$ (Where $\Re$ indicates the real part and $\Im$ indicates the imaginary part). The above series converges in the region of our interest which is $0 < \sigma < 1.$

To find the zeroes of $\zeta(s)$ we need to first substitute $\zeta(s) = 0$ and solve for $s$ and the sigmas and omegas.

That is

Solve $0= 1 – 1/2^s + 1/3^s -1/4^s + …$, for sigmas and omegas. Riemann hypothesized that the zeros will have their sigmas equal to 1/2 while the omegas are distinct. To this date, after 150 years, no one has any clue why sigma takes a single value of 1/2 in the critical strip $0 < \sigma < 1.$ Apart from the consequences I hope I explained it well. Wikipedia on Riemann Hypothesis is a good source for reading up.