Window to the World, calling for Wind of Wisdom, as common sense is a gift to each soul, as common environment is the inseparable planet, as common desire is to live in a better world.
Mathematicians make breakthrough in distribution of prime numbers 162 years after Riemann hypothesis, 3 centuries after Gauss’s estimate
Mathematicians Clear Hurdle in Quest to Decode Primes
Paul Nelson has solved the subconvexity problem, bringing mathematicians one step closer to understanding the Riemann hypothesis and the distribution of prime numbers.
It’s been 162 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers. Despite their best efforts, mathematicians have made very little progress on the Riemann hypothesis. But they have managed to make headway on simpler related problems.
In a paper posted in September, Paul Nelson of the Institute for Advanced Study has solved a version of the subconvexity problem, a kind of lighter-weight version of Riemann’s question. The proof is a significant achievement on its own and teases the possibility that even greater discoveries related to prime numbers may be in store.
The Riemann hypothesis and the subconvexity problem are important because prime numbers are the most fundamental — and most fundamentally mysterious — objects in mathematics. When you plot them on the number line, there appears to be no pattern to how they’re distributed. But in 1859 Riemann devised an object called the Riemann zeta function — a kind of infinite sum — which fueled a revolutionary approach that, if proved to work, would unlock the primes’ hidden structure. “It proves a result that a few years ago would have been regarded as science fiction,” said Valentin Blomer of the University of Bonn.
Hidden Structure of Prime Numbers Coming Into Focus
Just as molecules are composed of atoms, in math, every natural number can be broken down into its prime factors—those that are divisible only by themselves and 1. Mathematicians want to understand how primes are distributed along the number line, in the hope of revealing an organizing principle for the atoms of arithmetic.
For 165 years, mathematicians seeking that structure have focused on the Riemann hypothesis. Proving it would offer a Rosetta Stone for decoding the primes—as well as a $1 million award from the Clay Mathematics Institute. Now, in a preprint posted online on 31 May, Maynard and Larry Guth of the Massachusetts Institute of Technology have taken a step in this direction by ruling out certain exceptions to the Riemann hypothesis. The result is unlikely to win the cash prize, but it represents the first progress in decades on a major knot in math’s biggest unsolved problem, and it promises to spark new advances throughout number theory.
Predicting exactly where the next prime will show up on the number line is challenging, but describing the cumulative abundance of primes over large intervals is surprisingly straightforward. In the late 1700s, at the age of 16, German mathematician Carl Friedrich Gauss saw that the frequency of prime numbers seems to diminish as they get bigger and posited that they scale according to a simple formula: the number of primes less than or equal to X is roughly X divided by the natural logarithm of X. Gauss’s estimate has stood up impressively well. To the best mathematicians can tell, the actual number of primes bounces slightly above and below this curve up to infinity. That known primes follow such a simple formula so closely suggests the primes aren’t completely random; there must be some deep connections governing where they appear.
But mathematicians want to know exactly how well Gauss’s guess holds up—and why. In 1859, Bernhard Riemann, another renowned German mathematician, sought help from a different function, now called the Riemann zeta function. For inputs, the function takes complex numbers, which are a combination of real numbers and what mathematicians call “imaginary” ones: a normal number multiplied by the square root of –1. The function seems to capture the discrepancies between Gauss’s curve and the real distribution of primes. The places where Riemann’s function equals zero—referred to as zeta zeros—directly describe the fluctuating errors around Gauss’s curve.
Here, Riemann made his famous conjecture: ignoring certain trivial solutions for negative inputs, all the zeta zeros should exist for inputs where the real part is one-half. If his hypothesis is true, it means the seemingly random fluctuations in the abundance of primes are bounded, leaving no big clumps or gaps in their distribution along the number line. Any proof of the Riemann hypothesis would be a window into the secret clockwork governing the primes’ irregular pattern. It would offer a chance to “reverse-engineer the random number generator of the primes,” says Maksym Radziwill, a mathematician at Northwestern University.
