The Riemann Hypothesis Part 1

This post’s unsolved math problem is the Riemann Hypothesis. However, The Riemann Hypothesis is a rather complicated, and intricate problem. So the problem will be broken into two posts, one about zeta functions, and the second on the problem itself. This is done to give more background knowledge about the Riemann hypothesis so that from the reader perspective, the information can be consumed more easily. Within this post the goal is to get readers to understand zeta functions, and why they are connected to Riemann’s Hypothesis.

A zeta function, or also known as a Riemann-zeta function, can be first seen in the work of Leonard Euler. Euler used these in his work in the first half of the eighteenth century. A zeta function uses the lower case Greek letter zeta in its notation. It refer to a summation of the reciprocals of all whole numbers raised to the a value. So zeta (x) is equal to 1/(1^x) + 1/(2^x) + 1/(3^x) . . . all the way until infinity. Euler explored these functions and it resulted in some interesting results. For example, Euler evaluated zeta (2), or the same as 1 + 1/4 + 1/9 +1/16 . . . and the result is (pi^2)/6. That might seem a little weird, but at the same time it seems possible. One of the problems people have with zeta functions is that the function produces some strange results. If you evaluate zeta (0) or the equivalent of 1+1+1+1 . . . the answer is -1/2. That seems like an impossible result, because it is a linear growth, that is a non-convergent series, meaning that it does not approach anything but infinity. The hardest part about understanding zeta functions is wrapping your mind around this idea. There are plenty of debates out there that argue about these weird results, but it remains that this is a true result. The main argument is about the equal sign at the end. The argument is over whether the sum actually equals the result, or if it is an equivalent. Some people think that if you sum the zeta function all the way to infinity then you will have the result, but other people think that the result is just an equivalent to the answer, and the latter would be right.

It only seems right to mention Riemann when talking about Riemann zeta functions. Riemann simply expanded on the work of Euler. He expanded the Zeta function’s domain, and graphed it on the complex plane. That will be specifically covered in the next post, but it is good to know how Riemann is involved. The problem itself has to do the graphing of the function itself. Now when Riemann was working with zeta functions, he came across the most controversial/famous result from the zeta function. This being zeta (-1), or equivalent to 1 + 2 + 3 +4 . . . being the sum of all positive numbers. The solution to zeta (-1) is -1/12. It is quite a bizarre answer, but it is actually an important result in several branches of physics. This result was a big part in the explanation of the extra spacial dimensions in string theory. It also allowed for a solution to the Casimir force. It is not important to know about these physics terms, but the point is to show that these results are actually results in the real world.

The last point that needs to be stressed is that these sums are not directly equal to the answers given. As mentioned previously these results are more like equivalent to the sums. If you were to add all the positive numbers together you would get an infinite solution. So mathematicians look at this result, and say that result is useless. Instead of throwing all the math away, mathematicians say what if I could get a meaningful value out of this zeta function. It can be explained with a gold pan analogy. Imagine an infinite amount of dirt, and you use a sifting pan to look through the dirt. The zeta function result is like a nugget of gold that is found in this pile. Even though is it not equivalent to the pile it was the most meaningful thing in the pile. So no some of these zeta function result do not make sense, but they are a legitimate result that gives these summations a meaningful value.

Sources https://www.youtube.com/watch?v=d6c6uIyieoo
              https://www.youtube.com/watch?v=w-I6XTVZXww
              http://mathworld.wolfram.com/RiemannZetaFunction.html