Suppose a proof came out (and was verified by credible peer review) of the following statement:

There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$

where $T_0$ is totally ineffective. What interesting consequences would this partial result have?

Of course you could ask this sort of question for all kinds of weakenings/strengthenings/relatives of RH:

• Zero-density estimates (which already has its own question here)
• Density Hypothesis
• Lindelöf Hypothesis
• Generalized Riemann hypotheses for various L-functions
• Grand Lindelöf Hypothesis

But so far all the uses I have seen of $\zeta$ zeros has been in the strip $0<T<T_0$ and I wondered if that was convenience (where we've checked) or more than that.