The Riemann Hypothesis Part 2

This blog post is part two of the Riemann Hypothesis, the first part is also on this blog. The first part covered zeta functions, this was in order to give background knowledge allowing for this post to jump right into the problem without a long explanation, and allow for a fuller in depth explanation of the problem. However, I did not mention in the previous post that it is necessary to understand complex numbers, and the complex plane. Complex number were looked over because, that concept is taught to many kids in eighth grade math, so hopefully most of the audience already understands it. The goal of this post is to explain the actual problem, and secondly why the problem is important.

It was mentioned in the last post that zeta functions were a big part in understanding the Riemann Hypothesis. Also previously mentioned Euler worked with zeta functions. The difference between Euler’s and Riemann’s work was what they inputted into the zeta function. Euler focused on real numbers, Riemann on the other hand worked with complex numbers. Complex numbers are real number and imaginary number pairs, such as 3+4i, 3 being the real number, and 4i the imaginary.  When Riemann plugged in complex numbers into the zeta function it created a plane rather than a line. but after Riemann, it was understood that there was a whole extra dimension of values that could be put through the zeta function. The zeta function can now be visualized on a plane, the x axis being real numbers, and the y being imaginary numbers. Like all functions the zeta function has a domain, its domain is when x is great than 1. That includes all values both up, and down the imaginary axis, for x values greater than one. Then Riemann proved that the function has a rather nice piece of symmetry. Through analytical continuation Riemann prove you can extend the domain to all values except for x equals one. This leaves a strip between x=0 and x=1, this strip is referred to as the critical strip. Within the strip there is a critical line at x=1/2. That information may be hard to visualize so a graph will be included to help clear up any problems.

Now that the domain of the zeta function has been cleared up, the problem can be looked at. The million-dollar question is where does zeta (s) = 0? S is referring to the complex number as well as the real numbers. Another notation could be seen as s = x + iy. So the search is on for what point on this plane when plugged into the zeta function results in a zero. There are some freebees though, these are called trivial zeros. They are zero that we already know, and really do not care about. Such as every negative even number will result in a zero. The real question is, where are the zeros on the critical strip? Riemann hypothesized that all the zeros in the critical strip would lie on the critical line, at x=1/2. So far no one has found any zero that can disprove Riemann, and nobody has proven him either.

            The question might be asked, who cares? What does this arbitrary function have to with anything important? Something that was not mentioned in the zeta functions paper was that the zeta function could also be written in terms of prime numbers. So the position of these zeros allows for mathematicians to understand the position of prime numbers. For example, if the Riemann hypothesis was proven, it could be known how many prime numbers are there from 1 to a billion, or 1 to a trillion. This function also has ties to physics and quantum mechanics. This tells us that there is a lot of connections in math and physics that have not been made yet. If the function is linked to primes, and quantum mechanics, then how is quantum mechanics related to primes? There are lots of these connections that we do not understand at this moment, so if the Riemann Hypothesis is proven it opens the flood gates of math and physics to new ideas and connections. In doing so, unraveling mysteries and understandings of nature as a whole. That is why it is a million-dollar question, and why people are so interested in it.

        Sources https://www.youtube.com/watch?v=VTveQ1ndH1c
                      https://www.youtube.com/watch?v=sD0NjbwqlYw
                      http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html

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