Editing Riemann zeta function (section)

===The argument of the Riemann zeta function===
The function
:<math>S(t) = \frac{1}{\pi}\arg{\zeta\left(\tfrac12+it\right)}</math>
is called the [[complex argument|argument]] of the Riemann zeta function. Here {{math|arg ''ζ''({{sfrac|1|2}} + ''it'')}} is the increment of an arbitrary continuous branch of {{math|arg ''ζ''(''s'')}} along the broken line joining the points {{math|2}}, {{math|2 + ''it''}} and {{math|{{sfrac|1|2}} + ''it''}}.

There are some theorems on properties of the function {{math|''S''(''t'')}}. Among those results<ref>{{cite journal |first=A. A. |last=Karatsuba |title=Density theorem and the behavior of the argument of the Riemann zeta function |pages=448–449 |journal=Mat. Zametki |issue=60 |year=1996}}</ref><ref>{{cite journal |first=A. A. |last=Karatsuba |title=On the function {{math|''S''(''t'')}}| pages=27–56| journal= Izv. Ross. Akad. Nauk, Ser. Mat. |volume=60 |issue=5 |year=1996}}</ref> are the [[Mean value theorems for definite integrals|mean value theorems]] for {{math|''S''(''t'')}} and its first integral
:<math>S_1(t) = \int_0^t S(u) \, \mathrm{d}u</math>
on intervals of the real line, and also the theorem claiming that every interval {{math|(''T'', ''T'' + ''H'']}} for
:<math>H \ge T^{\frac{27}{82}+\varepsilon}</math>
contains at least
: <math> H\sqrt[3]{\ln T}e^{-c\sqrt{\ln\ln T}} </math>
points where the function {{math|''S''(''t'')}} changes sign. Earlier similar results were obtained by [[Atle Selberg]] for the case
:<math>H\ge T^{\frac12+\varepsilon}.</math>