Researchers have made what might be new headway toward a proof of the Riemann hypothesis, one of the most impenetrable problems in mathematics. The hypothesis, proposed 160 years ago, could help unravel the mysteries of prime numbers.
Mathematicians made the advance by tackling a related question about a group of expressions known as Jensen polynomials, they report May 21 in Proceedings of the National Academy of Sciences. But the conjecture is so difficult to verify that even this progress is not necessarily a sign that a solution is near (SN Online: 9/25/18).
At the heart of the Riemann hypothesis is an enigmatic mathematical entity known as the Riemann zeta function. It’s intimately connected to prime numbers — whole numbers that can’t be formed by multiplying two smaller numbers — and how they are distributed along the number line. The Riemann hypothesis suggests that the function’s value equals zero only at points that fall on a single line when the function is graphed, with the exception of certain obvious points. But, as the function has infinitely many of these “zeros,” this is not easy to confirm. The puzzle is considered so important and so difficult that there is a $1 million prize for a solution, offered up by the Clay Mathematics Institute.
But Jensen polynomials might be a key to unlocking the Riemann hypothesis. Mathematicians have previously shown that the Riemann hypothesis is true if all the Jensen polynomials associated with the Riemann zeta function have only zeros that are real, meaning the values for which the polynomial equals zero are not imaginary numbers — they don’t involve the square root of negative 1. But there are infinitely many of these Jensen polynomials.
Studying Jensen polynomials is one of a variety of strategies for attacking the Riemann hypothesis. The idea is more than 90 years old, and previous studies have proved that a small subset of the Jensen polynomials have real roots. But progress was slow, and efforts had stalled.
Now, mathematician Ken Ono and colleagues have shown that many of these polynomials indeed have real roots, satisfying a large chunk of what’s needed to prove the Riemann hypothesis.
“Any progress in any direction related to the Riemann hypothesis is fascinating,” says mathematician Dimitar Dimitrov of the State University of São Paulo. Dimitrov thought “it would be impossible that anyone will make any progress in this direction,” he says, “but they did.”
It’s hard to say whether this progress could eventually lead to a proof. “I am very reluctant to predict anything,” says mathematician George Andrews of Penn State, who was not involved with the study. Many strides have been made on the Riemann hypothesis in the past, but each advance has fallen short. However, with other major mathematical problems that were solved in recent decades, such as Fermat’s last theorem (SN: 11/5/94, p. 295), it wasn’t clear that the solution was imminent until it was in hand. “You never know when something is going to break.”
The result supports the prevailing viewpoint among mathematicians that the Riemann hypothesis is correct. “We’ve made a lot of progress that offers new evidence that the Riemann hypothesis should be true,” says Ono, of Emory University in Atlanta.
If the Riemann hypothesis is ultimately proved correct, it would not only illuminate the prime numbers, but would also immediately confirm many mathematical ideas that have been shown to be correct assuming the Riemann hypothesis is true.
In addition to its Riemann hypothesis implications, the new result also unveils some details of what’s known as the partition function, which counts the number of possible ways to create a number from the sum of positive whole numbers (SN: 6/17/00, p. 396). For example, the number 4 can be made in five different ways: 3+1, 2+2, 2+1+1, 1+1+1+1, or just the number 4 itself.
The result confirms an earlier proposition about the details of how that partition function grows with larger numbers. “That was an open question … for a long time,” Andrews says. The real prize would be proving the Riemann hypothesis, he notes. That will have to wait.