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).

Does this contradict something?

Even if it is true, a conditional proof probably will be hard yet welcome.

For Siegel $$Z$$ function on the critical line RH implies this for $$t$$ large enough.

Maybe can be generalized to $$\sigma \le \frac12$$.

Plot of a random interval:

• It seems that the zeros of the function you plotted are abscissas of inflexion points. – Sylvain JULIEN Jul 29 '13 at 11:17
• @SylvainJULIEN I am not sure they are really inflexion points, though they are very close to them. Here is a plot of the second derivative: s11.postimg.org/woe6fbzhv/re_zeta_0_123_inflexion.png The plot legend shows the function. – joro Jul 29 '13 at 12:03
• "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
• Note that under RH, as it has been proved for $\sigma = \frac12$ in the other thread, if ever there is a counterexample for $\sigma <\frac12$, this implies by continuity that there must be a $\sigma_0 < \frac12$ and $t$ such that $\Re \zeta(\sigma_0+i t)$ or $\Im \zeta(\sigma_0+i t)$ has a double zero. Not that this simplifies the thing, but I guess it is an interesting fact. – Wolfgang May 24 at 13:28