Wednesday, August 28, 2019

[xgivszps] Riemann Hypothesis probably true, based on observations of primes

There are many theorems conditional on the Riemann Hypothesis (even more the Generalized Riemann Hypothesis), or logically more strongly, many (some? are there any?) conditional theorems logically equivalent to the Riemann Hypothesis.  There are probably many more such conditional theorems that we haven't discovered yet.

Given that so many properties of prime numbers are conditional on RH, we conjecture that RH is true because if it isn't, we would have already noticed something weird going on with the primes.  In our universe where RH is true, the weirdness does not occur because of (possibly still unnoticed) properties of prime numbers conditional on RH, causing them to behave not weirdly.

We would have noticed something weird going on with primes because we regularly do many things with primes: various cryptography, prime searches, factoring projects.

Important instances of primes not behaving weirdly may still be unnoticed because humans are bad at detecting things not out of the ordinary.  Consider using AI trained on existing theorems conditional on RH to discover, or conjecture, more.

Is it true that we've sufficiently investigated (albeit unsystematically) the primes for weirdness?  Many proofs conditional on RH can (probably) be turned into computational investigations searching for indirect evidence disproving RH -- evidence suggesting the existence of a counterexample beyond where region where RH has been explicitly verified.  Have we done those computations?

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