What is known about the behavior of the argument of Riemann’s Zeta function on other verticals inside the critical strip apart from the critical line ? Are there any omega type theorems in this case, as is the case for $S(t)$ ? To be specific, for example, what is known about the density of sign changes of the argument of Zeta on the vertical $Re(z) = 0.8$ ?


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I'm not sure what sign changes are in this context, as there is no simple multiplier which forces the zeta function to be real on this strip. Instead the argument is going to be moving around in the circle.

However, for most versions of this question, a positive answer is going to follow from the result of Bagchi (exposited in Chapter 3 of An Introduction to Probabilistic Number Theory by Emmanuel Kowalski), who showed that the values of the Riemann zeta function on random short intervals of a vertical axis in the critical strip away from the critical line are equidistributed according to a certain measure on the space of holomorphic functions (and even the same thing for discs in the critical strip).

  • $\begingroup$ From the universality property of $log(\zeta)$ we know that the argument of Zeta on $Re(z)=0.8$ fluctuates wildly, but that doesn't answer my question about the density of sign changes @WillSawin $\endgroup$ Feb 5 at 2:22
  • $\begingroup$ @CristianDumitrescu Oh, I see what you mean. You're taking the argument, as a real number, and examining when it's sign changes. But doesn't the (probabilistic) universality imply that the density of points where the sign changes is linear? The probability the argument has a sign change in an interval is the probability a random function (according to a certain distribution) has a sign change in the interval which is positive. $\endgroup$
    – Will Sawin
    Feb 5 at 3:05

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