| ▲ | dist-epoch a day ago | |
Had same problem. One thing which is not always made clear: sound waves can be thought as a sum of sinusoidals of different frequencies (the Fourier Transform). The zeta functions does something similar: you add up the zeta functions for the non-trivial zeros, and the sum of them has jumps at the location of primes. Highly simplified, and also wrong, you don't get the actual primes, but an error correcting term to another prime estimation function. So in a way, the zeroes of the zeta function encode "the frequencies of the primes" (in the Fourier sense) | ||
| ▲ | jesuslop 21 hours ago | parent [-] | |
Apropos, Wikipedia says at a picture: "(Left) The von Mangoldt function, approximated by zeta zero waves.(Right) The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes." [1]. [2] explains the details very gently, one can skip parts. So this is a machine with input a function very close to prime counts, and outputs info about zeta zeros. And the machine is Fourier transform. [1] https://en.wikipedia.org/wiki/Von_Mangoldt_function [2] Prime numbers and the Riemann hypothesis. Mazur, Stein | ||