Prime numbers are sometimes called math’s “atoms” because they can be divided by only themselves and 1. For two millennia, mathematicians have wondered if the prime numbers are truly random, or if some unknown pattern underlies their ordering. Recently number theorists have proposed several surprising conjectures on prime patterns—in particular, probabilistic patterns that show up in large groups of the mathematical atoms.
The patterns in the primes trace back to an 1859 hypothesis involving the legendary Riemann zeta function. Mathematician Bernhard Riemann derived a function that counts the number of primes up to a number x. It includes three main ingredients: a smooth estimate, a set of corrective terms coming from the Riemann zeta function, and a small error term.
Much has been written about the Riemann zeta function, but the most important thing to know is that it provides a correction to the smooth estimate. To do so, it takes on a wavy pattern, sometimes raising the count, sometimes lowering it. These corrective oscillations are determined by the locations of the zeros of the Riemann zeta function. In fact, the celebrated Riemann hypothesis claims that all such zeros lie on a “critical line” where the real part equals 1⁄2.
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The zeros intrigue mathematicians for two reasons. First, they imply that the zeta function is encoding as-yet-unknown information about the primes. Second, they suggest that the spacing of the primes, despite irregularities, is as orderly as possible; smaller fluctuations would contradict the density of the primes.
Taken together, this means the error in Riemann’s prime counting formula is as minimal as possible.
The hypothesis has been verified all the way…
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