by Bogdan Teodorescu / Scientists

It’s the birthday month of Bernhard Riemann, who might very well be one of the most brilliant mathematicians Earth has ever had. From contributions such as proposing the world’s most challenging math problem (still unsolved, and you could earn 1 million dollars if you solve it) to laying the grounds for General Relativity, this man changed completely the way we look at mathematics.

## Early life of Bernhard Riemann

On 17 September 1826, Bernhard Riemann was born in Germany to a pastor father and a mother who died early in his life. He was poor and known as a shy person, being more of an introvert. So in 1846, he started studying philology and theology to help his family’s material situation by becoming a pastor, such as his father.

This was not for long; as soon as his father saved enough money, he sent him to the University of Gottingen to study mathematics. This also happened in 1846, when Riemann was 19. He studied under the well-known Carl Friedrich Gauss, who appreciated Bernhard Riemann a lot, stating even that he has *gloriously fertile originality, which was said as a comment to his Ph.D.* thesis, called “Basics for a general theory of functions of a variable complex size.” Gauss was also the one who supported his relentless mind into his strange ideas of non-euclidean geometry.

Although Bernhard Riemann studied mathematics with Carl Gauss, he was not enrolled in theology. However, at Gauss’ insistence, he transferred to the University of Berlin in 1847 to study mathematics. Many distinguished mathematicians of the times were teaching there, such as Jacobi, Steiner, or Dirichlet. In 1849, Bernhard Riemann returned to Gottingen, where his heart was.

## The Riemann integral

In high school, many of you have heard about definite integrals. Well, I call them so too, but they are actually Riemann integrals because Bernhard Riemann was the first to rigorously define what they are. He defined Riemann sums and taught us that integrals are Riemann sums of infinitely (that’s how we think of it) small rectangles, which in the end give us the most accurate value for the area under a curved graph.

## A new geometry and the foundations of general relativity

The non-euclidean geometry I’ve been speaking about is also an amazingly important contribution to science. It introduced such concepts as the metric tensor, geodesics, or the curvature tensor. Basically, this geometry studies curved surfaces, and by this, you can think of spheres or cylinders, for example. One absolutely convincing difference between the two geometries, which also makes it clearer that there is a need for another set of laws and rules, is that in the Euclidean geometry we all know, there is that axiom which says that between two parallel lines, there is always the same distance, where on a sphere, for example, there are no parallel lines!

Another interesting thing is that if you draw two points on a paper and want to find the shortest route from one to another, you will see that there is only one good option, drawing a perfect line between them. There is no other ‘shortest route’, whereas on a sphere, if you want, for example, to find the shortest route from the ‘North Pole’ to the ‘South Pole’ (think of Earth), you can find a lot of routes with the same distance!

*Mass bends the fabric of spacetime*

Einstein’s theory of relativity deals with space-time, a mathematical model combining the three dimensions of space that we can see with a fourth, the dimension of time. This space is similar to a manifold, the mathematical objects that Bernhard Riemann was studying. Unfortunately, Einstein was unaware of the mathematical apparatus that would describe the gravitational field’s properties, so he sought help from acquaintances in the scientific world, notably Hermann Minkowski, one of his teachers. Minkowski explained the geometry in special relativity, but the problems were harder to solve regarding general relativity.

Finally, Grossmann, Einstein’s best friend, made the creator of general relativity aware of Bernhard Riemann’s mathematics and the concept of tensors. Think of a vector, and the way they work in one dimension, with two components. A tensor is basically an extended vector, which for a quantity acting in 4 dimensions, would need 16 components. That was called a metric tensor, and on paper, it is actually a matrix.

With the help of Bernhard Riemann’s ideas, Einstein explained two things in one: how gravitational forces make things move and how more or less matter changes gravitational forces, which led him to the idea that gravity is not necessarily a force, but it is the magnitude of change in the gravitational field, or in the space-time continuum.

## The 1-million dollar problem by Bernhard Riemann

The Riemann hypothesis states: *“the real part of every non-trivial zeros of the Riemann zeta function is ½..*” What does that mean? First, the Riemann zeta function looks like this:

The zero of this function is when you basically find the x for which the y is 0. You know that if you want to graph a function f(x) = y in an x-y system (cartesian coordinates), the y axis stands for the value of the function. In the case of our function, finding the zero of the function means finding the value for s for which ζ(s) = 0.

An imaginary number is just like a real number, an irrational number, or a natural number, but this is a *complex *number. It has an imaginary part, written as *i*, and can be written as *a + bi*. Here, *a *is the real part of this number, by definition.

The function has zeros for -2, -4, -6, and so on, but these are trivial zeros, and we look for non-trivial zeros. These all lie within a critical strip between 0 and 1, as shown above. Strange enough, the zeros don’t just lie in the portion we see, but on that exact line, the critical line. So the real part of all the non-trivial zeros is 1/2. That’s the hypothesis. Interestingly there is no way to prove the hypothesis by brute calculation, so there must be something else, and scientists are still looking for it.

In 1866, Bernhard Riemann died in Italy after he fled Germany because of the armies of Hanover and Prussia, where he was also buried in Biganzolo. His tombstone once again proved his dedication to God, stating,

Here rests in God

Georg Friedrich Bernhard Riemann

Professor in Göttingen

born in Breselenz, 17 September 1826

died in Selasca, 20 July 1866

For those who love God, all things must work together for the best.

He was a devout Christian, and he saw his life as a mathematician as another way to serve God, which is very true, and it is beautiful to think like that. After all, mathematicians decipher the language this whole Universe speaks. God may have spoken to us through different people who have spoken some things that are now written in our books. Still, above all, God might have spoken to us through all the beautiful things that exist in the Universe, which cannot be explained otherwise through the language of mathematics.