Math: Physicists Attack Math’s $1,000,000 Question

Quanta Magazine said:

Prime numbers, the indivisible atoms of arithmetic, seem to be strewn haphazardly along the number line, starting with 2, 3, 5, 7, 11, 13, 17 and continuing without pattern ad infinitum. But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function.

The Riemann zeta function takes inputs that can be complex numbers — meaning they have both “real” and “imaginary” components — and yields other numbers as outputs. For certain complex-valued inputs, the function returns an output of zero; these inputs are the “nontrivial zeros” of the zeta function. Riemann discovered a formula for calculating the number of primes up to any given cutoff by summing over a sequence of these zeros. The formula also gave a way of measuring the fluctuations of the primes around their typical spacing — how much larger or smaller a given prime was when compared with what might be expected.

However, Riemann knew that his formula would be valid only if the zeros of the zeta function satisfied a certain property: Their real parts all had to equal ½. Otherwise the formula made no sense. Riemann calculated the first few nontrivial zeros of the zeta function and confirmed that their real parts were equal to ½. The calculation supported his hypothesis that all zeros had this property, and thus that the spacing of all prime numbers followed from his function. But he noted that “without doubt it would be desirable to have a rigorous proof of this proposition.”

A century and a half later, proving the Riemann hypothesis remains arguably the most important unsolved problem in pure mathematics — one whose solution would fetch a $1 million Millennium Prize from the Clay Mathematics Institute. Conversely, as the number theorist Enrico Bombieri wrote in his description of the problem, “the failure of the Riemann hypothesis would create havoc in the distribution of prime numbers.”

As mathematicians have attacked the hypothesis from every angle, the problem has also migrated to physics. Since the 1940s, intriguing hints have arisen of a connection between the zeros of the zeta function and quantum mechanics. For instance, researchers found that the spacing of the zeros exhibits the same statistical pattern as the spectra of atomic energy levels. In 1999, the mathematical physicists Michael Berry and Jonathan Keating, building on an earlier conjecture of David Hilbert and George Pólya, conjectured that there exists a quantum system (that is, a system with a position and a momentum that are related by Heisenberg’s uncertainty principle) whose energy levels exactly correspond to the nontrivial zeros of the Riemann zeta function. Each of these energy levels, En, corresponds to a zero of the form Zn = ½ + iEn, which has a real part equal to ½ and an imaginary part formed by multiplying En by the imaginary number i.

If such a quantum system existed, this would automatically imply the Riemann hypothesis. The reason is that energy levels of quantum systems are always real numbers (as opposed to imaginary), since energy is a physically measurable quantity. And since the En’s are purely real, they become purely imaginary when multiplied by i in the formula for the corresponding Zn’s. There is never a case where an imaginary part of En is multiplied by i, canceling out its imaginary property and rendering it real, so that it then contributes to the real component of Zn and changes it from ½ to something else. Since energy levels are always real, the real parts of the zeros of the zeta function would always be ½, and the Riemann hypothesis would therefore be true.

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