the Riemann zeta function
/ REE-mahn ZAY-tuh /
The Riemann zeta function is, at first sight, an innocent infinite sum: zeta(s) = sum over n >= 1 of 1/n^s = 1 + 1/2^s + 1/3^s + 1/4^s + ... . You met its special values already without the name — zeta(2) = 1 + 1/4 + 1/9 + ... = pi^2/6 is the Basel sum, and the harmonic series is the divergent case s = 1. The novelty is to treat the exponent s as a complex variable. For Re s > 1 the series converges absolutely and defines a holomorphic function; that convergent series is the gateway, not the whole story.
The deep step, due to Riemann, is analytic continuation: although the series only converges for Re s > 1, the function it defines extends UNIQUELY to a holomorphic function on the entire complex plane except for a single simple pole at s = 1 (with residue 1, reflecting the harmonic series' logarithmic divergence). Once continued, zeta has values everywhere: zeta(0) = -1/2, zeta(-1) = -1/12, zeta(-2) = 0, and so on. These are honest analytic values of the continued function, NOT sums of the divergent series; the famous '1 + 2 + 3 + ... = -1/12' is a loose shorthand for zeta(-1) = -1/12 and is not literal summation.
Why does this one function command such attention? Because of the Euler product, zeta secretly encodes the prime numbers, and its complex zeros control how the primes are distributed. The prime number theorem, the prime counting function's fine structure, and the still-open Riemann hypothesis all live inside zeta. It is the founding object of analytic number theory and arguably the most studied function in mathematics. The whole subject of this field is, in a sense, the study of why an exponent written as a complex number unlocks the integers.
For Re s > 1 the series is concrete: zeta(3) = 1 + 1/8 + 1/27 + 1/64 + ... = 1.2020569..., Apery's constant (proved irrational in 1978). At s = 2 the sum is exactly pi^2/6. But evaluate the CONTINUED function at s = -1 and you get zeta(-1) = -1/12 — a value the original series, which there reads 1 + 2 + 3 + ..., could never produce by summation.
zeta(s) = sum 1/n^s only for Re s > 1; values like zeta(-1) = -1/12 belong to the continued function, not the divergent series.
zeta has exactly one singularity — a simple pole at s = 1; it is holomorphic everywhere else. The 'trivial zeros' at s = -2, -4, -6, ... are genuine zeros of the continued function, not artefacts of the divergent series.