The Riemann Hypothesis, Explained

Riemann applied analytic continuation to extend Euler’s zeta function from real numbers s > 1 to the entire complex plane. This technique from complex analysis allows functions defined on limited domains to be extended while preserving their analytic properties.

Riemann identified zeros of the extended zeta function. Trivial zeros occur at negative even integers (-2, -4, -6, …). Non-trivial zeros lie in a vertical strip in the complex plane. Computing projects have verified over 10 trillion non-trivial zeros.

Carl Friedrich Gauss studied prime number distribution in the late 1700s, calculating prime tables up to 3 million. The prime counting function π(x) formalizes tracking how many primes appear up to any number x.

Bernhard Riemann conjectured in his 1859 paper that all non-trivial zeros of the zeta function lie on the critical line. This hypothesis carries a $1 million Clay Mathematics Institute Millennium Prize. Riemann proved infinitely many zeros exist in the critical strip but could not prove all lie on the line.

Leonhard Euler defined the zeta function ζ(s) as the sum of 1 over positive integers raised to power s. Euler solved ζ(2) = π²/6 and calculated ζ(4), ζ(6), ζ(8). Bernhard Riemann later extended this function to complex numbers in 1859.