What Is the Riemann Hypothesis?

Alex Kontorovich says the Riemann hypothesis predicts that every wave behind the prime numbers has exactly the same amplitude. The Riemann zeta function takes complex numbers in and sends complex numbers out, and it governs how the primes are distributed. A zero is a complex input that makes the output equal zero, the origin in the complex plane.

People want to know how the primes are spread through the natural numbers. Riemann's breakthrough was signal processing on the primes, the same move a microphone makes with sound. A voice box makes pressure waves, Fourier analysis turns the waves into amplitudes and frequencies, and a speaker rebuilds the voice from the stored numbers. The zeta function runs an analysis on the primes, thought of as air molecules wiggling, and the analysis can recover the primes. The frequencies and the amplitudes correspond to the zeros of the zeta function.

The zeta function holds the information. Superposing the waves, which he calls Riemann waves, recovers the distribution of the primes. Under the Riemann hypothesis, the amplitudes are all exactly the same, like an orchestra playing the music of the primes with every instrument at the same volume.