One infinite sum quietly encodes the pattern of every prime number
Add one, a half squared, a third squared and so on forever, and you have touched the Riemann zeta function. Euler found it was secretly built from the primes. Riemann then pushed it into complex numbers and left behind a conjecture many mathematicians rank as pure mathematics' greatest open problem.
The zeta function, written with the Greek letter of that name, takes a complex number s and sums the reciprocals of every positive whole number raised to the power s. That sum only settles to a finite value when the real part of s exceeds 1. Elsewhere the function is defined by analytic continuation, a way of extending it smoothly across the complex plane. The result behaves well everywhere except at s equal to 1, where the sum becomes the divergent harmonic series and the function has a single pole.
Leonhard Euler was the pioneer. In 1737 he proved that the infinite sum equals an infinite product taken over all primes, an identity now called the Euler product, whose proof needs only geometric series and the fact that every integer factors uniquely into primes. In 1740 he studied the series for positive whole-number inputs, and Chebyshev later widened it to any s with real part above 1. Euler also computed its values at the even positive integers, the first of which solved the Basel problem, and at the negative integers, where the answers are rational numbers.
The prime connection has sharp consequences. Because the harmonic series diverges, the product formula implies there are infinitely many primes and, more strongly, that the reciprocals of the primes also add up to infinity. The same product gives the chance that several randomly chosen integers share no common factor.
Bernhard Riemann's 1859 paper on counting primes below a given size moved the function into complex variables, proved its functional equation linking s with 1 minus s, and tied its zeros to how primes are spread out. It also stated the Riemann hypothesis about where the nontrivial zeros lie. Separately, Roger Apéry showed in 1979 that the value at 3 is irrational, and that number now bears his name.
Source: Riemann zeta function
