Based on limited numerical evidence, I am inclined to suspect that there is always zero of $\Re \zeta(1/2+it)$ between consecutive local extrema of $\Re \zeta(1/2+it)$ (and the same for $\Im \zeta(1/2+i t)$).

There are very short intervals having two zeros.

For Siegel $Z$ function, RH implies this for $t$ large enough.

Is it true (maybe conditionally)?

 

Counterexamples? (Please check for 2 zeros in a short interval).

Based on limited numerical evidence, I am inclined to suspect that there is always zero of $\Re \zeta(1/2+it)$ between consecutive local extrema of $\Re \zeta(1/2+it)$ (and the same for $\Im \zeta(1/2+i t)$).

There are very short intervals having two zeros.

For Siegel $Z$ function, RH implies this for $t$ large enough.

Is it true (maybe conditionally)?

Counterexamples? (Please check for 2 zeros in a short interval).