Author at ‘The Secrets Of The Universe’, I am a science student from Romania. I am also the founder of Astronomy Hub, an organization for popularizing astronomy and astrophysics. I love reading philosophy and literature, enjoy classical rock, blues, and watch movies.
The Riemann Hypothesis. Probably the best-known unsolved problem in mathematics. Well, just as in every other scientific field, we still have unsolved mysteries and discoveries for which scientists give their whole lives. One interesting feature of mathematics is that problems tend to stretch for very great amounts of time. Fermat’s Last Theorem (also called the Fermat conjecture), another significant problem in mathematics, was stated in 1637 by Pierre de Fermat and proved in, guess what, 1994, by Andrew Wiles. That means 358 years of mathematicians showing up trying to solve the problem.
Now the Riemann Hypothesis is not that old, dating from 1859 when Bernhard Riemann stated it. It may be the hardest problem to solve in mathematics right now, and that could be because of its apparent simplicity. As you will see, you will have no problem understanding its statement.
The Riemann-Zeta Function
This basically has to do with a function, which I will show you in a bit. First, we know what a function is, right? A function is a law, a relationship, that describes what happens with a variable (or a constant, if you want). From a different perspective, you can say that a function deals with how a mathematical quantity is related to another. I am taking a random function example, and I am coming up with this: f(x) = 2x + 1. It means that if we give the value 4 to x, the function f (x) will be equal to f (4) = 2*4+1 = 9.
This is a random example, which probably doesn’t mean that much in our everyday lives (although I don’t guarantee that, it may mean something). What I want to say is that we encounter functions every day in our lives because nature can be described with the help of functions, and everything around us basically works after a mathematical law.
Now the function we have here, the Riemann zeta function, looks a bit more complicated than my example. It looks like this
This is a sum of an infinite sequence; that is why we have those “…” to show that the sum goes on. Many functions look like this, and many of them have critical applications in our lives, such as harmonic analysis or thermodynamics. If you wish to calculate the Riemann Zeta function for any integer number, you need to plug in the value. Let’s say you wish to calculate ζ(2). Substitute the value s=2 in the above function and you’ll get an infinite series 1/12 + 1/22 + 1/32 + 1/42 + … and so on. This series converges to a value of π2/6. Don’t think where did π come from. For now, remember that the series converges to a particular value whenever s >= 1.
Getting back to our Riemann zeta function, one wants to find the “zero” of the function. What’s that? You’ll understand very quickly: the “zero” of the function f (x) = 2x+1 is -½. Why? Because for x = -½, f (x) (actually, f(-½) ) will be equal to 0. So in the case of the Riemann zeta function, its zero is any value of s for which ζ(s) = 0.
The last thing you must know before understanding this million-dollar problem is the concept of complex numbers. I am sure most of you are already familiar with it but let’s have a quick look at it.
We all know that the square of two integers is always a positive number: 2*2 = (-2)*(-2) = 4. But mathematicians realized it’s important to have a number whose square is negative. They couldn’t find any such number, so they made it and called it an imaginary number. The denoted this imaginary number by i, the square root of -1. So i2 = -1.
The set of complex numbers includes the set of real numbers, the set of rational numbers, the set of integer numbers, and natural numbers. Of course, besides containing all those sub-sets, it has something new. We now write every complex number as a + bi, where a and b are some real numbers, and i is the number’s imaginary part. The real part of the number is a in this definition.
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The Riemann Hypothesis
The Riemann Hypothesis says this: the real part of every non-trivial zeros of the Riemann zeta function is ½.
I know it’s a bit difficult to absorb in one go! See, by analytic continuation, the Riemann Zeta function becomes zero for all the negative integers: -2, -4,-6, etc. These are the trivial zeroes. We aren’t interested in the trivial ones. We are interested in the non-trivial zeroes that all lie within a critical strip between 0 and 1, as shown above. Peculiarly, the zeroes not just lie in the shaded portion but on a critical line. The real part of all the non-trivial zeroes is 1/2. That’s the hypothesis.
Using computers, we have checked up to a trillion points, and all of them lie on the critical line. We just need one out of infinite points to reject the Riemann hypothesis. But brute computation will never be able to prove/reject the hypothesis. We need rigorous mathematical proof of the Riemann hypothesis: one that goes for all the numbers up to infinity. Prove the Riemann hypothesis, and you’ll become immortal in the world of mathematics. You may have guessed it by now, conceptually, it is more difficult, and obviously, solving the problem involves almost all the mathematics we know by now (and some of the maths we don’t know yet).
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A Million-Dollar problem
Although the problem has been stated, what, 162 years ago (check my maths if you want), it can be said that it came into the spotlight in 1900, when a renowned mathematician, David Hilbert, listed it on his shortlist of 10 greatest unsolved mathematical problems, at the Second International Congress of Mathematics, at Paris. Since then, Stephen Smale, an American mathematician, updated the list, positioning the problem in the 1st place among the most important unsolved problems.
If you want to know, solving this problem could mean that you don’t have to do anything else for the rest of your life. The problem is also listed among the Millenium Problems, which would guarantee 1 million dollars to their solver as one of the seven problems. Don’t get me wrong, 1 million for each problem, not 1 million for all of them. That’s a lot of money. Thankfully, nobody does that for the sake of the money (actually, even mathematicians who won a fair amount of money from their discoveries most of the time donated them).