The counting function for the number of complex zeros of the Riemann $\zeta$ function up to height $T$ is well known. I am looking for a reference that gives essentially analogous counting functions for the number of zeros of the real part of $\zeta$ (on the critical line), and likewise the counting function for the imaginary part. Thanks, Andrey
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2$\begingroup$ Maybe you're not anticipating what kind of answer there'll be: zeros of the real (or imaginary) part will be ("real") curves in the complex plane... not isolated points... (And, of course, the intersections of these two batches of curves are the isolated points which are the zeros of zeta...) $\endgroup$– paul garrettCommented Jun 18, 2023 at 3:57
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$\begingroup$ By the term "essentially" I basically meant their zeros on the critical line (not that other vertical lines would not be of interest). $\endgroup$– AndreyFCommented Jun 18, 2023 at 4:06
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$\begingroup$ Ah, ok! I did wonder. :) Maybe modify your question accordingly? $\endgroup$– paul garrettCommented Jun 18, 2023 at 4:07
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