Is there an absolute mimumum distance between zeros of the Riemann zeta function?

Question

I believe that the nontrivial zeros of the Riemann zeta function get increasingly spaced out the further they are from the real axis. This suggests that there is probably an absolute minimum distance between any pair of zeros of the zeta function. Is this true?

If there's no known minimum, is there a positive infimum of the set of distances between all pairs of zeros of the zeta function? Or a positive distance such that only a finite set of pairs are closer together than this distance (i.e. a positive lim inf of the sequence of distances between consecutive zeros)?

Does the answer to any of those questions change if we only consider the zeros on the critical line, or if we assume the Riemann hypothesis? Are any of the propositions above suspected to be true, even if they aren't known to be true?

Answer 1

Your belief that the nontrivial zeta zeros get sparser as you increase the imaginary part is false. In fact (see here), it is known that $N(T)$, the number of (nontrivial) zeros of the Riemann zeta function with imaginary part between $0$ and $T$, grows like $\Theta(T\log T)$. This implies that there must be two zeros with imaginary parts differing by $\Theta(1/\log T)$ and imaginary parts less than $T$.

Coupled with the fact that a positive proportion of zeta zeros lie on the line $\Re s=1/2$, this implies that there are pairs of zeros arbitrarily close together.

Answer 2

Taking @CarlSchildkraut's answer a bit further: in fact, Montgomery pair-correlation conjecture (for example) would imply that even the renormalized gaps (divided by $1/\log T$ at height $T$) have inf $0$. Currently, there's some upper bound for that inf that is well below $1$...

That is, the more interesting discussions use renormalized gaps...