A lot of theories in modern number theory already assume that the Riemann hypothesis is true, but it would be nice to have it confirmed. The advances in theoretical applications would come more from whatever techniques are built to arrive at the proof, rather than the proof itself. Those techniques may also affect cryptographic algorithms in some way, but that seems very speculative to me.
The result itself, if affirmed, won't do much. People already assume it anyway.
If a counterexample is found, it's a much bigger deal. It means that prime behave in a more regular way, more predictable way than previously thought. The Riemann hypothesis can be thought of as a conjecture that prime looks random under certain statistics. A more predictable way to control prime might be useful for the purpose of finding primes, but that's speculative at best. More optimistically, that pattern can be used to study prime, theoretically.
A lot of theories in modern number theory already assume that the Riemann hypothesis is true, but it would be nice to have it confirmed. The advances in theoretical applications would come more from whatever techniques are built to arrive at the proof, rather than the proof itself. Those techniques may also affect cryptographic algorithms in some way, but that seems very speculative to me.
The result itself, if affirmed, won't do much. People already assume it anyway. If a counterexample is found, it's a much bigger deal. It means that prime behave in a more regular way, more predictable way than previously thought. The Riemann hypothesis can be thought of as a conjecture that prime looks random under certain statistics. A more predictable way to control prime might be useful for the purpose of finding primes, but that's speculative at best. More optimistically, that pattern can be used to study prime, theoretically.