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?
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asked Feb 11, 2020 at 12:34
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1$\begingroup$ This can be rephrased as follows: the Hilbert transform of the real part is close to the derivative. Or, in yet another form: the Fourier transform of $t \mapsto \Re \zeta(1/2+i t)$ is concentrated around $\{-1, 0, 1\}$. This indeed is the case, and can be quantified in some sense; see, for example, this answer. $\endgroup$– Mateusz KwaśnickiCommented Feb 11, 2020 at 13:17
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1$\begingroup$ Well, I should have rather written "this indeed may be the case, and possibly could be quantified". I do not claim I can quantify this at that moment. $\endgroup$– Mateusz KwaśnickiCommented Feb 11, 2020 at 13:23
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(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.
answered Feb 11, 2020 at 19:37
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$\begingroup$ Thank you for your answer. Denoting respectively the derivative of the real part and the imaginary part by $Zr'$ and $Zi$, could one expect to get a result along the lines of $\int_{x_{0}}^{x}(Zr'(t)-Zi(t))dt\ll_{\varepsilon}x^{\varepsilon}$ for some $x_{0}>1$ and all $\varepsilon>0$? $\endgroup$ Commented Feb 11, 2020 at 19:56
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$\begingroup$ I doubt this is any more regular than the integral of $\Im z(t)$, but honestly I know next to nothing about the Riemann zeta function. $\endgroup$ Commented Feb 11, 2020 at 20:16
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$\begingroup$ Do you think my question in my previous comment should be asked separately? $\endgroup$ Commented Feb 11, 2020 at 20:29
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$\begingroup$ @SylvainJULIEN: Again, I do not know, you may ask someone more knowledgeable. $\endgroup$ Commented Feb 11, 2020 at 20:42
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1$\begingroup$ There certainly does seem to be some sort of correlation: consider
ListPlot@Table[{Re[I Zeta'[1./2 + I t]], Im[Zeta[1./2 + I t]]}, {t, 0, 800}]andListPlot@Table[{Re[I Zeta'[1./2 + I t]], Im[Zeta[1./2 + I t]]}, {t, 700, 800, 0.1}]. $\endgroup$ Commented Feb 12, 2020 at 0:21