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The argument of the Riemann zeta function

The function

is called the argument of the Riemann zeta function. Here arg ζ (1 / 2 + it) is the increment of an arbitrary continuous branch of arg ζ (s) along the broken line joining the points 2, 2 + it and 1 / 2 + it.

There are some theorems on properties of the function S (t). Among those results are the mean value theorems for S (t) and its first integral

on intervals of the real line, and also the theorem claiming that every interval (T, T + H ] for

contains at least

points where the function S (t) changes sign. Earlier similar results were obtained by Atle Selberg for the case