# Naive conjecture about zeros and local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$) for $0 \le \sigma \le \frac12$

Based on limited numerical evidence, I suspect this conjecture.

Conjecture: Fix $0 \le \sigma \le \frac12$ and let $t > 0$. Between consecutive local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$), there is always a zero of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$).

Verification for several random $\sigma$ and $0 < t < 30000$ and for a few random intervals didn't show any counterexamples.

For $\sigma > \frac12$ it is false and on the other hand this appears counterintuitive to me.

Counterexamples? (please check for closely spaced zeros that might look like a single local minimum on a large plot).

For Siegel $Z$ function on the critical line RH implies this for $t$ large enough.
Maybe can be generalized to $\sigma \le \frac12$.
"For $\sigma>1/2$ it is false and on the other hand this appears counterintuitive to me." This may be related to the fact that RH is equivalent to $\zeta^\prime(s)$ has no zeros in $\sigma\le 1/2$. But $\zeta^\prime(s)$ does have zeros in $\sigma>1/2$. –  Stopple Aug 6 '13 at 17:31