Riemann zeta function
Number theory
The function ζ(s) = ∑ 1/nˢ summed over positive integers n, defined by that series where it converges and extended to the rest of the complex plane by analytic continuation.
What is usually left out
Its connection to the primes is the Euler product: ζ(s) also equals the product over all primes p of 1/(1 − p⁻ˢ). That identity is why a question about a complex-analytic function is a question about prime distribution. Note that "zeta function" names a whole family — Selberg, Ihara, dynamical — most of which have nothing to do with primes.
Related terms
In the atlas
- Riemann hypothesis — Number theory, open