Saturday, January 12, 2013

A Dumbing Down of the Riemann Hypothesis

Today is my first post on math in the news. I recently came across this article on the Riemann Hypothesis, which I had planned to talk about in India, but didn't get a chance to. Let me give a brief background and then I will share the article.

Back in 2000, Clay Mathematics Institute of Providence, Rhode Island announced the Millennium Prizes, which consisted of seven problems that had been stumping mathematicians for a long time. They set aside a million US dollars for any person who solved one of the problems.

I find it interesting just on its own that you can become wealthy as a mathematician. Other than the Nobel Economics Prize, math has its own way of getting a million dollars.

One of the more popular of these problems is the Riemann Hypothesis, which I wanted to talk about today. I will try to explain here what the Riemann Hypothesis is (it is a difficult concept, but online sources complicate it drastically), and then show the article.

First off, you might remember the number i, which is the square root of -1. This is not a normal variable that can just replace anything you want; it is always the square root of -1. You may have heard in math class the term "real number." A number that has just an i in it are imaginary numbers, like 2i or -5i.

A little over a year ago, I did a post about the complex plane. This takes our horizontal number line from first grade and makes it our x-axis. It then takes these imaginary numbers and makes those the intervals of the y-axis.

A point on the x-axis is a real number, represented with the letter a. A point on the y-axis is an imaginary number, represented with the term bi. A point that is just floating around somewhere not on one of these lines is a complex number. You can write it with the expression a + bi, with a being the number it lines up with on the x-axis and bi being the number it lines up with on the y-axis.

Say you had to take the equation y = x^3 - 2 and start plugging in values for x (replacing the x with a number and then figuring out what it equals). Most people would start plugging in real numbers like 0, 1, 2, 3, -1, -2, -3, and so on. However, this Riemann Hypothesis requires us to open up our minds a little bit. Rather than just plugging real numbers into equations, we have to plug complex numbers into equations.

The Riemann Zeta Function is the equation we are plugging these numbers into (a little side-note: the Riemann Zeta Function has a 2π in it for any tauists). For the Riemann Hypothesis, it is concerned to find when this equation equals zero. This is called the zero of the equation.

The real numbers that are zeros of this equation are called the trivial zeros. They are equal to -2, -4, -6, -8, and so on. The complex numbers that are zeros of this equation are called the non-trivial zeros. As far as we know, these non-trivial zeros all have different b values in our a + bi, but the a value always seems to be 1/2.

The Riemann Hypothesis is simply asking the question is there a non-trivial zero of the Riemann Zeta Function whose a value is not equal to 1/2. Imagine winning a million dollars after submitting a hundred page paper that answers just a yes or no question.

It seems like proving either side would be extremely difficult. This article gives a nice explanation of how they are going about proving the yes side of it.

The article brings up a very good way to do it. If you find just one time where the a value is not 1/2 and it is complex, then the statement is proven. So, Yuri Matiyasevich decided to turn the supercomputers on and start cranking out values. They have not found any values without the 1/2, but they also cannot mathematically prove that it always is the 1/2, and that is where the dispute lies.

Let me finish by saying what I find cool about the Riemann Hypothesis. So what if some weird function might have a consistency with its x-intercepts? But there is a very practical and interesting connection.

You may notice how with the prime numbers, there really isn't any relation between them. I mean, the Fibonacci numbers are the sum of the two before it, the powers of two are one more than the sum of all the ones before it, the triangulars are the sum of the natural numbers, the squares are the sum of the odd natural numbers, but the primes have no relation like that. I have always wondered why that is, or if there was one.

If the Riemann Hypothesis gets solved, it will shine a light on the distribution of prime numbers. We will be able to see if they do have a pattern or if there is no pattern. Some might find the technological, algebraic, or financial parts of this problem interesting, but I think the practical aspect is really cool.


  1. Referring to the last paragraph of yours, I did encounter such a proof. Good or less good is a matter for the mathematician to decide, how convincing it seems to be. Then again I encountered proofs which were accepted by the mathematical community, yet proved to be false. I do believe that the abovementioned theory is not of this kind, that it may be verifiable. If so, then one theorem by Littlewood is false. I do not consider that a problem since Erdos did prove Littlewood wrong back in 1948. This would be just a dejavu. It is in the interest of maths to uncover half truths or even false statements, just to make it consistent. The book by Feliksiak : the symphony of primes just fits that category. But it comes awfully hard in detailed knowledge. Maybe that is a drawback.

  2. Recently I came across a review of the book mentioned above at:

    I find it quite impressive, equally well as the book.