Original version of This story Appeared in Quanta Magazine.
Mathematicians sometimes try to tackle problems head-on, but sometimes they go off on tangents, especially when the mathematical stakes are as high as the Riemann Hypothesis, which has a $1 million prize offered by the Clay Mathematics Institute for its solution. If proven, the Riemann Hypothesis would give mathematicians greater certainty about the distribution of prime numbers and also imply many other results, making it arguably the most important unsolved problem in mathematics.
Mathematicians don’t know how to prove the Riemann hypothesis at all. But they can get useful results simply by showing that there are a limited number of exceptions to it. “In many cases, this can be as good as the Riemann hypothesis itself,” says James Maynard of the University of Oxford. “This allows us to get similar results for prime numbers.”
In groundbreaking results published online in May, Maynard and Larry Gass of the Massachusetts Institute of Technology set a new upper limit on the number of exceptions of a particular kind, finally beating a record set more than 80 years ago. “This is a sensational result,” said Henryk Iwaniec of Rutgers University. “It’s very, very, very difficult. But it’s a gem.”
The new proof automatically leads to a better approximation of how many primes there are in a short interval on the number line and provides many other insights into the behavior of primes.
Careful Avoidance
The Riemann hypothesis is a claim about 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 can grow to any size as more and more terms are added. Mathematicians would say that the series diverges. But if we summarize it instead like this:
1 + 1/22 + 1/32 + 1/42 + 1/52 + ⋯ = 1 + 1/4 + 1/9+ 1/16 + 1/25 +⋯
Get Pi2/6, or approximately 1.64. Riemann’s surprisingly powerful idea was to convert the series into a function,
ζ(s) = 1 + 1/2s + 1/3s + 1/4s + 1/5s + ….
Therefore, ξ(1) is infinity, but ξ(2) = π2/6.
Things get really interesting s It is a complex number, consisting of two parts. The “real” part is an everyday number, and the “imaginary” part is an everyday number multiplied by the square root of -1 (or IComplex numbers can be plotted on a plane, with the real part being X-axis and imaginary part Yeah-axis. Here, for example, 3 + 4I.
Graph: Marc Beran, Quanta Magazine
