‘Gem’ of a Proof Breaks 80-Year-Old Record, Offers New Insights Into Prime Numbers

The archetypal version of this story appeared in Quanta Magazine.

Sometimes mathematicians effort to tackle a occupation caput on, and sometimes they travel astatine it sideways. That’s particularly existent erstwhile the mathematical stakes are high, arsenic with the Riemann hypothesis, whose solution comes with a $1 cardinal reward from the Clay Mathematics Institute. Its impervious would springiness mathematicians overmuch deeper certainty astir however premier numbers are distributed, portion besides implying a big of different consequences—making it arguably the astir important unfastened question successful math.

Mathematicians person nary thought however to beryllium the Riemann hypothesis. But they tin inactive get utile results conscionable by showing that the fig of imaginable exceptions to it is limited. “In galore cases, that tin beryllium arsenic bully arsenic the Riemann proposal itself,” said James Maynard of the University of Oxford. “We tin get akin results astir premier numbers from this.”

In a breakthrough result posted online successful May, Maynard and Larry Guth of the Massachusetts Institute of Technology established a caller headdress connected the fig of exceptions of a peculiar type, yet beating a grounds that had been acceptable much than 80 years earlier. “It’s a sensational result,” said Henryk Iwaniec of Rutgers University. “It’s very, very, precise hard. But it’s a gem.”

The caller impervious automatically leads to amended approximations of however galore primes beryllium successful abbreviated intervals connected the fig line, and stands to connection galore different insights into however primes behave.

A Careful Sidestep

The Riemann proposal is simply a connection astir a cardinal look successful fig mentation called the Riemann zeta function. The zeta (ζ) relation is simply a generalization of a straightforward sum:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ⋯.

This bid volition go arbitrarily ample arsenic much and much presumption are added to it—mathematicians accidental that it diverges. But if alternatively you were to sum up

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

you would get π²/6, oregon astir 1.64. Riemann’s amazingly almighty thought was to crook a bid similar this into a function, similar so:

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 truly absorbing erstwhile you fto s beryllium a analyzable number, which has 2 parts: a “real” part, which is an mundane number, and an “imaginary” part, which is an mundane fig multiplied by the quadrate basal of −1 (or i, arsenic mathematicians constitute it). Complex numbers tin beryllium plotted connected a plane, with the existent portion connected the x-axis and the imaginary portion connected the y-axis. Here, for example, is 3 + 4i.