TITLE: Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?
QUESTION [5 upvotes]: Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection points of the former. Can this be made precise? For example, is there a canonical notion of distance between those two functions that attains only small values?

REPLY [3 votes]: (An extended comment.) The derivative of the real part does not really seem to be close to the imaginary part, as seen in the following picture generated by Mathematica:

Code: Plot[{Re[I Zeta'[1/2 + I t]], Im[Zeta[1/2 + I t]]}, {t, 0, 80}]
The corresponding zeroes of the two functions indeed seem to be reasonably close to each other. This is no surprise, however: $\zeta(\tfrac{1}{2} + i t)$ essentially circles around (mostly in the right half-plane). In each "circle" the distance between a maximum or minimum of the real part and the corresponding zero of an imaginary part is roughly as large as the distance froth the "center" to the real axis. And most "circles" seem to be centered near the real axis.