This reminds me of the duality principle that Andrew Wiles used to prove Fermat’s Last Theorem with modular elliptical curves.
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.” … (Quanta Magazine)