Number theory: can the Riemann Hypothesis be proved to solve the pattern of prime numbers?

An image of numbers to illustrate the Riemann Hypothesis concept and the mystery of prime numbers
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What is the Riemann Hypothesis, and can it be proved to solve the underlying pattern of prime numbers?

According to Emory Health Sciences, the Riemann Hypothesis is a vehicle for understanding one of the greatest mysteries in number theory, the pattern underlying prime numbers.

The mystery underlying prime numbers

Prime numbers are defined simply as any number greater than one with no positive divisors except one and itself. However, the distribution of prime numbers remains hidden.

Ken Ono, a number theorist at Emory University and co-author of the paper, explained: “It’s well known that there are infinitely many prime numbers, but they become rare, even by the time you get to the 100s. In fact, out of the first 100,000 numbers, only 9,592 are prime numbers, or roughly 9.5 percent. And they rapidly become rarer from there. The probability of picking a number at random and having it be prime is zero. It almost never happens.”

The Riemann Hypothesis

The new paper supports Ono’s theory that “an old, abandoned approach to the Riemann Hypothesis should not have been forgotten.”

This refers to the hypothesis by the mathematician Bernhard Riemann in 1859. According to Emory Health Sciences, “In mathematical terms, the Riemann Hypothesis is the assertion that all of the nontrivial zeros of the Zeta function have real part ½.”

Ono commented: “His hypothesis is a mouthful, but Riemann’s motivation was simple. He wanted to count prime numbers.”

The new paper

The paper builds on the work of the twentieth century mathematicians Johan Jensen and George Pólya, giving a method to calculate the Jensen-Pólya polynomials – a formulation of the hypothesis – all at once rather than one at a time.

While it does not fully prove the hypothesis, consequences include previously open assertions which are known to follow from the Riemann Hypothesis, as well as some proofs of conjectures in other fields.

Ono added: “The beauty of our proof is its simplicity. We don’t invent any new techniques or use any new objects in mathematics. We provide a new view of the Riemann Hypothesis. Any reasonably advanced mathematician can check our proof. It doesn’t take an expert in number theory.”

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