Riemann Hypothesis

        The Riemann hypothesis is one of the millennium problems that were established in 2000 by the Clay Mathematics Institute. These seven problems were put forth by the Clay Institute as the most important unsolved mathematics problems at the turn of the millennium. If you solve one of the millennium problems, the Clay Institute will award you a million dollars. So far only one of the problems has been solved, the Poincare conjecture. The Poincare conjecture is a problem in topology, a field that studies different kinds of surfaces without caring about their dimensions. The problem says that every smooth three dimensional shape can be simplified into a sphere. It was solved by Grigori Perelman in 2003, but he denied the money and the nobel prize that he was offered for his discovery. The Riemann hypothesis is a conjecture about the Riemann zeta function () named after Bernhard Riemann. This says that all the solutions for  lie on a line on the complex plane where the real part is . If this sounds like gibberish, don’t worry, it took me a while to understand what that means, but I hope to explain it to you as well.

        Bernhard Riemann born in 1826 in Breselenz, Hanover. He made a lot of contributions to geometry that helped Einstein with his theory of relativity. He also contributed a lot to the study of complex functions. The imaginary number  is the , called imaginary because there is no real number that is the square root of a negative number. A complex number is a combination of a real number and a complex number in the form  where  and  are real numbers and  is the square root of negative one. This means that  is the solution to . From this description you can see that imaginary numbers are just real numbers multiplied by . Complex functions can just be normal functions, but the variable is not limited to real numbers anymore, but extended to complex numbers.

        The Riemann zeta function is one of these complex functions. The riemann function is This function creates an infinite series that follows that pattern forever. There are two important types of infinite series: convergent and divergent. If a series converges it means that it gets closer to a specific point as the number of terms gets closer to infinity. If a series diverges it means that it doesn’t get closer to a specific number. For example, the series converges to the number two. This means that after the pattern is repeated for an infinite amount of time the sum will be 2. Another example is the series . This series diverges which means that no matter how many terms you add it will never get closer to a number. For this reason, only real numbers greater than one converge in the Riemann zeta function. This is also true when you add complex numbers into the mix. Only numbers greater than 1 converge including numbers like  because it is larger than . When complex numbers become involved there is another way to analyse the function. This is called complex analysis and it allows for the points that converge to be stretched to the points that aren’t and assign a value.

        In complex analysis, specifically analytical continuation, a graph of a power series in the complex plane can be extended past its domain. This means that a graph of an infinite series that involves powers that increase term by term. The zeta function creates a power series. The domain of the zeta function is just the area of the complex plane which creates a convergent series when any value from that area is put in as . The domain for the zeta function is any complex number greater than 1. To extend this domain, you need to find another function with a different domain. These two domains must intersect and the functions must be equal at these intersection points. This means that the domain of the other function must also stop at the line at 1. This is why even with analytic continuation is still undefined. The zeta function is actually very symmetrical in a certain sense which makes the analytic continuation also symmetrical. By using this method strange things can happen with the values assigned to the values of  that are normally outside of the domain of the function. One of these examples is . It seems very counterintuitive that the sum of all natural numbers is negative and a fraction. For this reason you can’t really say that it equals . Instead you would say that is its assigned value according to analytic continuation of the function.

        With normal functions sometimes there are no values of  where the function is equal to 0 (e.g. ). When you allow for complex values of  there are always solutions to . Obviously, this means that there must be numbers for which . There are two different kinds of zeros for this function right now: trivial and nontrivial. The trivial zeros are the ones that are easy to predict and the nontrivial zeros are not. This is because there is an equation called the functional equation that allows you to find out what the value of the zeta function at any variable. This function is  Since  is always zero if  is an integer, this will equal zero for every even number. This would make the value of the function zero at all even numbers except for that little  beside it. This is called the gamma function, it’s the same kind of function as the zeta function, but it doesn’t follow the same pattern for outputs. In the gamma function, if  is even and positive, the result of  will be something called a simple pole. A simple pole is a value of a function like the gamma and zeta functions that acts kind of like . When you multiply a simple pole with a simple zero, which is what you get from the sin function, they cancel each other out. This is  way the trivial zeros are only negative positive integers.

The nontrivial zeros are a bit more complicated. It has been proven that they are all between the area of domain where the real portion of the complex number is greater than 0, but less than 1, but it is not easy to find where they all are. The Riemann hypothesis states that they are all on a line in the domain where the real part of the complex number is . If this hypothesis is correct it provides insight on the distribution of prime numbers, meaning how common prime numbers are between any two numbers. This may not have many applications in everyday life, but the Riemann hypothesis and the zeta function itself have lots of applications in the world of physics. For example  shows up in string theory and curiously when the assigned value is used in place of the infinite summation of all natural numbers, the problems work as they should.

I hope I have provided you enough information about Riemann and his zeta function to

help you understand how huge the scope of this problem is and why you would receive a million dollars for solving it.  

Sources

Gray, J. J. (2017, February 29). Bernhard Riemann. In Encyclopedia Britannica. Retrieved May 31, 2017, from https://www.britannica.com/biography/Bernhard-Riemann

Weisstein, E. W. (n.d.). Analytic Continuation. In Wolfram Mathworld. Retrieved May 31, 2017, from http://mathworld.wolfram.com/AnalyticContinuation.html

Rowland, T. (n.d.). Simple Pole. In Wolfram Mathworld. Retrieved May 31, 2017, from http://mathworld.wolfram.com/SimplePole.html