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On the distribution of values of the argument of the Riemann zeta-function. (English) Zbl 1428.11146

The argument function is defined as \(S(t):=\frac{1}{\pi}\arg \zeta(\frac{1}{2} + it)\) for \(t>0\) and \(\frac{1}{2} + t\) not a zero of of the Riemann zeta function \(\zeta\). But if \(\frac{1}{2} + \gamma\) is a zero of \(\zeta\), for \(t=\gamma\), we define \(S(t):=S(t+0)\). The main goal of this paper is the investigation of the distribution of positive and negative values of \(S(t)\).
Main results.
Theorem 1.
Suppose that \(0 < \varepsilon < 10^{-3}\) is an arbitrary small fixed constant, \(T \geq T_0(\varepsilon)\), \(T^{c+\varepsilon} \leq H \leq T\), with \(c = \frac{27}{82}\). Then for any real numbers \(a < b\), \[ \mathrm{mes}\left\{t \in \left[T, T+H\right] : a < \frac{\pi\sqrt{2} S(t)}{\sqrt{\log\log T}} \leq b\right\} = \frac{H}{\sqrt{2\pi}}\left(\int^b_a e^{-v^2 /2}\mathrm{d} v + \mathcal O\left(\frac{\log_3 T}{\varepsilon\sqrt{\log_2 T}}\right)\right), \] where the \(\mathcal O\)-constant is absolute and \(\mathrm{mes}\{\cdot\}\) denotes the measure.
Theorem 2. In the same conditions as in Theorem 1, \[ \mathrm{mes}\left\{ t\in \left[T, T+H\right] : S(t) > 0\right\} = \frac{H}{2} + \mathcal O \left(\frac{H \log_3 T}{\varepsilon\sqrt{\log_2 T}}\right), \] where the \(\mathcal O\)-constant is absolute.
A more complex theorem and its corollary round up the results list.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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References:

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