Читать книгу The Number Mysteries: A Mathematical Odyssey through Everyday Life - Marcus Sautoy du - Страница 27

The million-dollar question is about the nature of these dice: are the dice fair or not? Are the dice distributing the primes fairly through the universe of numbers, or are there regions where they are biased, sometimes giving too many primes, sometimes too few? The name of this problem is the Riemann Hypothesis.

Bernhard Riemann was a student of Gauss’s in the German city of Göttingen. He developed some very sophisticated mathematics which allows us to understand how these prime number dice are distributing the primes. Using something called a zeta function, special numbers called imaginary numbers and a fearsome amount of analysis, Riemann worked out the mathematics that controls the fall of these dice. He believed from his analysis that the dice would be fair, but he couldn’t prove it. To prove the Riemann Hypothesis, that is what you have to do.

Another way to interpret the Riemann Hypothesis is to compare the prime numbers to molecules of gas in a room. You may not know at any one instance where each molecule is, but the physics says that the molecules will be fairly evenly distributed around the room. There won’t be one corner with a concentration of molecules, and at another corner a complete vacuum. The Riemann Hypothesis would have the same implication for the primes. It doesn’t really help us to say where each particular prime can be found, but it does guarantee that they are distributed in a fair but random way through the universe of numbers. That kind of guarantee is often enough for mathematicians to be able to navigate the universe of numbers with a sufficient degree of confidence. However, until the million dollars is won, we’ll never be certain quite what the primes are doing as we count our way further into the never-ending reaches of the mathematical cosmos.