Complexity in the Simplicity - Part I Riemann Hypothesis

From the very first day that humans started looking into the secrets of Mother Nature, mathematics became his ultimate companion. Throughout several centuries, being the key tool for revealing most of world’s mysteries, mathematics have become the biggest companions of modern science.

Mathematics is not something given by god. Mathematics is not something created overnight. It was the ultimate result of the utter dedication and continuous sacrifices of brilliant minds which ever existed. Throughout the history of mathematics there are a lots of remarkable milestones. It is undoubted that every milestone was a game changing point in mathematics as well as in the modern science.

As it is mentioned, mathematics is shaped up by the people who had the highest IQ as well as genius thinking ability. But there are some unsolved mysteries in mathematics which were able to keep most of those brilliant minds busy throughout these centuries. As a fact some of those mysteries have histories of more than 200 years. Some of those mysteries were solved by people and most of those solutions have opened new paths in the library of mathematics. But still there are some mysteries in mathematics which can lead to an entire transformation of modern science and technologies as we know them. So let’s dig deep into those mysteries in the world of mathematics.

This will be a series of articles about the most complex problems in mathematics.

“Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” — Albert Einstein

As Einstein said these articles are for the people who love imagination. Even though we are about to dive deep into some complex areas in mathematics I assure you, this won’t be a high school math class. This will reveal the beauty of unsolved mysteries which is hidden behind the complex equations and Greek letters.

Unsolved Mysteries - I - Riemann Hypothesis

What is this Riemann Hypothesis and who is this Riemann? First things first. Riemann Hypothesis is one of the most complex mathematical problems that exists today. This was first came to the picture in 1859. Since then world’s genius, brilliant minds have struggled with this more than a century. In 2000 Clay Mathematics Institute announced this as a million dollar question. If someone is able to provide a valid proof for this mathematical problem, he will be awarded with this million dollar prize.

What is this question which is worthy million dollars? It’s all about a function. What is a function in mathematics? It is a simple statement which gives a value for a given number. As an example let’s consider f(y) = 2 + y. so when y = 1, f(1) = 2+1 = 3, f(1) = 3; when y=2, f(2) = 2+2 = 4, f(2) = 4; It is simple as it is. So this problem is also connected to a function. That function name is “Zeta” and it is also very much like other functions.

This is also just a function. Only difference of it is, it will go up to infinity without a finite boundary. When we give a value for “s” it will give us a number. Hence ζ(2) has a value, ζ(3) has a value.

In 18th Century Leonhard Euler one of the forefathers of modern mathematics found those values and he came up with interesting answers. Those answers were really breathtaking and they led Riemann to a hypothesis which today worth million dollars. (As a person who love mathematics I personally don’t believe assigning bunch of dollars for a priceless mathematical question is a good idea. It ruins the pure values of mathematics.) These are the answers when different numbers were substituted for Zeta function.

As we can see these infinite sums (Which is called in mathematics as infinite series) give us a definite value. Isn’t it amazing? Let’s think a bit. If we take a calculator or even a very high powerful computer can we do the addition manually? In our life time? In the entire life time of universe? NO we can’t because there is no end. It will go forever. But mathematics shows us the final answer. Simply that’s the power behind mathematics. Most of the mind blowing inventions and creations that we experience in our day today life are designed thanks to the power of expressions that mathematics has.

Coming back to the discussion, Even though this Zeta function gives definite values for most of the substitutions it has one weakness. When “s =1” this beautiful function fails to provide a definite answer so when “s =1” the sum becomes infinite. That means the total goes to infinity.

This great work of Euler was investigated by a young brilliant mind and he expanded the domain of this function to a new level. If you carefully look at previous example you would see that always “s” is substituted by positive values only. This young fellow started his investigation to expand the domain of the function. (Domain - Range of values that can be substituted to a function). What is the new Domain? The new domain includes complex numbers. Let’s keep our main discussion bit aside and look at what are these complex numbers.

This concept of complex numbers can be easily explained using a famous question. The square root of 9 equals to 3, the square root of 1 equals to 1, so what is the square root of -1? This question was the dawn of another branch of mathematics. Mathematicians declare a number called “i” and the value of the “i” is declare as “i² = -1”. Weird isn’t it? But that weird solution made a tremendous change. This changed the number line to a number plane.

This is the number line.All these numbers are called real numbers and from both sides it extends up to infinity. Fractions lie between those numbers. As an example half lie between zero and one.

This is the number plane. All these real number are in the real axis. There is another axis shown in this image named “Imaginary Axis”. This Imaginary axis contains imaginary numbers like 1i , 2i, 3i etc. In other words it can be explained as 1√-1, 2√-1,3√-1 etc. Since mathematicians declared this notation as Imaginary Axis, the previous number line became a number plane. As shown in the image each point in the plane can be represent by a number. So this weird definition gave a very large playground for our mathematician “Riemann” to test this Zeta function.

As one of the greatest mathematicians in history of mathematics Bernhard Riemann proved this entire number place has definite values for zeta function except at a specific point. Such kind of points are very special for a function and those are named as singularity points.

As this image shows Riemann was able to show that this Zeta function gives a well defined value in anywhere in the number plane except in 1.

Here comes the million dollars question. Once Riemann extended Euler’s work he was interested in particular behavior of the function. At which values this function becomes zero? Riemann came up with an awesome answer and that awesome answer finally created a problem which made most of the genius minds clueless. So brace yourselves. Here comes the Riemann Hypothesis.

After playing with complex analysis and fellow mathematicians’ works Riemann explained that there are only two types of zeros. One type is known as Trivial Zeros. When the “s” in “ζ(s)” becomes -2, -4, -6 or any other negative even integer this “ζ(s)” function becomes zero. So those zeros are explained as Trivial Zeros. All other values for “s” which makes “ζ(s) = 0 ” are know as non-Trivial Zeros.

Riemann Hypothesis - The real part of every non-trivial zero of the Riemann zeta function is 1/2.

So this is what Riemann predicts. He was able to prove that all non trivial Zeros exist between 0 and 1 in the number plane. That strip is known as the critical strip. But he said actually all the non trivial zeros only lie on the line of 1/2 and no where else. (The critical line)

So there are two ways of winning this million dollars

1. You have to prove this Riemann Hypothesis. Have to come up with a valid mathematical proof.
2. You have to show any value in this blue strip which is not in the critical line that makes “ζ(s)” zero !!!!

Second one looks much easy. But trust me, there are millions of attempts that have been made even with the aid of high end mathematical computing devices to find a non trivial zero which doesn’t lie on the critical line. But so far no luck. In the other hand thousands of non trivial zeros have been found on the critical line. Most of the mathematicians believe what Riemann claimed in his Hypothesis is true and currently we don’t have the knowledge to explore that.

So that’s it. That is the million dollar question. But why this finding non-Trivial Zeros worth million dollars? Why is it that much significant? Because if someone proves this Hypothesis, that may lead to an era where modern security becomes useless. Not just one ,but almost all of them. Because most of our modern security techniques including web security uses key base techniques. The complexity of hacking these key base techniques lies on the complexity of finding the prime numbers in mathematics. In other words if someone comes up with an algorithm which can run within a quarantine time period to find all the primes less than a given number, that will be the end of the key base encryption as well as key base security. Riemann Hypothesis is one of the keys to such kind of algorithm.

In mathematics the branch which deals with prime numbers and distributions is called Number theory. Riemann Hypothesis can open a new area of number theory only if it’s proved. Now more than 100 years have passed after this question is raised. But no one has come up with even a closer answer for this problem. So why don’t you a give try? May be you will be able to change the destiny of Mathematics as well as yours.