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August 23, 2007



Yes. It is unfortunate how hard it is to explain RH. I'd be great if it was like the one you mentioned by Fermat.

Anyways, the idea is that you have a function, "the zeta function" (put some echo here), defined on the complex plane (this is, the function takes complex numbers and gives you complex numbers). This "zeta function" is an "extension" of a real infinite series (one of those infinite sums Ramanujan loved) but I guess we can skip that detail by now.

The point is that if you do z(-2) or z(-4) or z(-6) (and so on, any negative even number), then you get 0. That's more or less easy to see once you know the formula for the zeta function. When they say "trivial zeros", they mean these guys ("the negative even numbers"). However, it was clear the zero function had more zeros (the so-called non-trivial ones) and some properties were known about them.

Parenthesis: To explain these properties, it is probably good to remember a little bit about the complex numbers. A complex number is something of the form a+ib where a and b are real numbers and i is that "unexistent" square root of -1 some of us meet in highschool some in college. We can represent the complex numbers in the cartesian plane assigning a+ib to the pair (a,b). Given a complex number a+ib, the Real Part (Re) of it is a and the Imaginary Part (Im) is b.

Ok, back to RH, the known properties of the non-trivial zeroes of the zeta function were two:

1. The non-trivial zeroes were all located in the vertical strip with Real Part between 0 and 1. This is, comples numbers a+ib where 0 2. If we had a non-trivial zero and we look for the symmetrical complex number (in the cartesian plane) with respect to the line Re(x)=1/2 (The vertical line of complex numbers whose real part is 1/2), then we would find ANOTHER non-trivial zero.

These people studing the zeta function, however, noticed something else: Most of the non-trivial zeros they knew were actually ON the line Re(x)=1/2.

Riemann Hypothesis claims that ALL the non trivial zeros are on the Re(x)=1/2 line. That's it.

The connection between the RH and the distribution of primes is another story and it should probably be told some other time. (Actually, that's the part that's really hard to explain in elementary terms.)

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