The Riemann Hypothesis.

In the year 2000, the Clay Mathematics Institute compiled a list of the seven most challenging problems in all of mathematics and offered a million dollars to anyone who could either prove or disprove one of these conjectures. One of these Millennium Prize problems was the Riemann Hypothesis. It's one of the most complicated questions ever conceived by mankind and solving it has often been called the most difficult way to win a million dollars. So let's get into it. The Riemann Hypothesis was a conjecture posed by the great German mathematician Bernard Riemann in 1859. It lies in the field of complex analysis, which is basically the study of functions of complex variables. However, it also has huge implications in a field of mathematics called number theory, which deals with the prime numbers. The problem has to do with something called the Riemann zeta function. You see, Riemann was interested in a particular infinite series, one over one to the power s plus one over two to the power s plus one over three to the power s, etc. For different integer values of s, he called this his zeta function named after the Greek letter. For some values of s, for example, s equals two or four, the function was convergent and had some pretty interesting outputs. A few other people had been interested in the series before, so it was nothing special. But what Riemann did was ingenious. He said, instead of integers, let's consider plugging in complex numbers as values of s. Through some very advanced mathematics, he managed to come up with an elegant formula for the zeta function that extended the domain or input values to include complex numbers as well. You don't really have to worry about the formulas because they're extremely complicated. And the methods used to come up with this are a whole nother video in and of itself. But what's important is that by writing in these forms, values that were previously divergent suddenly became convergent. The details are a bit obscure, but essentially Riemann use something called analytic continuation to find another formula for his function that showcased a symmetry between the values of zeta of s and zeta of one minus s. This allowed him to define values that previously blew up the function. The only catch was that zeta of one, which corresponds to one plus one half plus one third, etc, was still undefined. Okay, so what's the million dollar question? Well, Riemann wanted to know what values of s made zeta of s equals zero. Doesn't seem so hard when you look at it like that. Some of these zeros were obvious by plugging into the equation, the negative even integers, negative two, negative four, negative six, etc. But Riemann didn't care about these trivial zeros. He was interested in the non trivial zeros. He managed to prove that all the non trivial zeros would lie in a critical strip in between the vertical lines at zero and one on the complex plane, but couldn't make any further progress. However, he did notice that all of the non trivial zeros seem to be concentrated around the vertical line at one half. Riemann's hypothesis was that all the non trivial zeros of the zeta function are complex numbers with a real part of one half. Basically, this means that when you graph the function on the complex plane, all of the zeros will lie on the vertical line at one half. So far through computer search, we've been able to find trillions of zeros that fit the statement, but we still don't have a rigorous proof. Okay, so why exactly is this esoteric question one of the most important problems in all of modern mathematics? Well, it turns out that the hypothesis is very closely related to number theory, a field of pure mathematics. Number theory deals with the study of the integers, the most fundamental type of numbers. A huge part of number theory concerns the prime numbers, numbers that are only divisible by one and themselves. Mathematicians are as crazy about primes as physicists are about fundamental particles, and they've been studying them for millennia, attempting to find different patterns in their distribution. Riemann discovered that a zeta function could also be written in a very strange way as a product over the primes. This was a revolutionary idea, because what he showed was Needless to say, it was a huge accomplishment. So overall, what does this teach us? Well, aside from all that we've learned from studying the Riemann hypothesis, I feel that the problem itself is a comment on human nature. I think the fact that we're even able to conceive something as intricately complicated as this is remarkable, and says something about humans and how far we've come as a species. Not too long ago, we used to think the earth was flat, and now look at us. We've still got a long way to go to solve this problem, but I think with a little bit of creative thinking, anything's possible.