The Riemann Hypothesis: why zeros on a line should matter to anyone who cares about primes

The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function lie on the critical line with real part 1/2. This is not obviously connected to prime numbers until you understand that the zeta function encodes the distribution of primes in its zeros — and that the error term in the prime number theorem is controlled by how far those zeros might be from the critical line.

If the hypothesis is true, primes are 'as regular as they can be' given their apparent randomness. If it's false, there are zeros off the line, and the prime distribution has irregularities we've never detected but couldn't rule out.

Over 10 trillion zeros have been computed and verified to lie on the critical line. Not one exception has been found. The hypothesis is almost certainly true. We cannot prove it.

What I find compelling about this for independent researchers isn't the prospect of solving it — that would require machinery far beyond any individual — but what it reveals about the nature of mathematical truth. We can be empirically confident in a mathematical claim with no counterexamples after enormous computational search, and that confidence is categorically different from proof.

What does it mean to 'know' something in mathematics? Is numerical evidence epistemically different in mathematics than in science?