The ‘gem’ of proof breaks an 80-year-old record and provides new insights into prime numbers.

Original version ~ Of This story appeared in Quanta Magazine.

Mathematicians sometimes try to solve problems head-on, and sometimes they approach them sideways, especially when the stakes are as high as the Riemann hypothesis. The solution to the Riemann hypothesis is being offered as a $1 million reward by the Clay Mathematics Institute. The proof gives mathematicians much greater confidence in how prime numbers are distributed, while also implying many other results. It is perhaps the most important unsolved problem in mathematics.

Mathematicians have no idea how to prove the Riemann hypothesis. But they can still get useful results by showing that there are a limited number of possible exceptions to the Riemann hypothesis. “In many cases, it may be as good as the Riemann hypothesis itself,” he said. James Maynard “We can get similar results from this for prime numbers,” said Oxford University.

in Groundbreaking results Maynard and posted online in May Larry Guss A Massachusetts Institute of Technology professor has set a new upper limit on the number of exceptions of a certain type, finally breaking a record set more than 80 years ago. “It’s a remarkable result,” he said. Henryk Iwaniec “Very, very, very difficult. But it’s a gem,” said Rutgers University.

A new proof automatically derives a better approximation of how many primes exist within a short interval of the number line, and will provide many other insights into how primes work.

Careful avoidance

The Riemann hypothesis is a statement of a central formula in number theory, called the Riemann zeta function. The zeta (ζ) function is a generalization of simple sums.

1 + 1/2 + 1/3 + 1/4 + 1/5 + ⋯.

This series will grow arbitrarily as more terms are added. Mathematicians say that this diverges. But instead, if we sum up the following:

1 + 1/2² + 1/3² + 1/4² + 1/5² + ⋯ = 1 + 1/4 + 1/9+ 1/16 + 1/25 +⋯

you will get π²/6, or about 1.64. Riemann’s amazingly powerful idea was to turn a series like this into a function like this:

ζ(S) = 1 + 1/2^S + 1/3^S + 1/4^S + 1/5^S + ⋯.

So ζ(1) is infinite, but ζ(2) = π²/6.

Things get really interesting if you let them. S It is a complex number consisting of two parts: a “real” part, which is an everyday number, and an “imaginary” part, which is an everyday number multiplied by the square root of -1. me(As mathematicians write) Complex numbers can be displayed on a plane, and their real part can be displayed on a plane. X-Axis and imaginary part why-axis. For example, here it is 3 + 4.me.