Editing Riemann zeta function (section)

==Various properties==
For sums involving the zeta function at integer and [[half-integer]] values, see [[rational zeta series]].

===Reciprocal===
The reciprocal of the zeta function may be expressed as a [[Dirichlet series]] over the [[Möbius function]] {{math|''μ''(''n'')}}:
:<math>\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}</math>
for every complex number {{mvar|s}} with real part greater than 1. There are a number of similar relations involving various well-known [[multiplicative function]]s; these are given in the article on the [[Dirichlet series]].

<!--The paragraph below needs to be explained better; we need a section on RH equivalents. -->
The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of {{mvar|s}} is greater than {{sfrac|1|2}}.

===Universality===
The critical strip of the Riemann zeta function has the remarkable property of '''universality'''. This [[zeta function universality]] states that there exists some location on the critical strip that approximates any [[holomorphic function]] arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.<ref>{{cite journal|last=Voronin|first=S. M.|date=1975|title=Theorem on the Universality of the Riemann Zeta Function|journal=Izv. Akad. Nauk SSSR, Ser. Matem.|volume=39|pages=475–486}} Reprinted in ''Math. USSR Izv.'' (1975) '''9''': 443–445.</ref> More recent work has included [[Zeta function universality#Effective universality|effective]] versions of Voronin's theorem<ref>{{ cite journal |author1=Ramūnas Garunkštis |author2=Antanas Laurinčikas |author3=Kohji Matsumoto
 |author4=Jörn Steuding |author5=Rasa Steuding |title=Effective uniform approximation by the Riemann zeta-function |journal=Publicacions Matemàtiques  |date=2010 |volume=54 |issue=1 |pages=209–219 |doi=10.5565/PUBLMAT_54110_12 |jstor=43736941 |url=http://ddd.uab.cat/record/52304 }}</ref> and [[Zeta function universality#Universality of other zeta functions|extending]] it to [[Dirichlet L-function]]s.<ref>{{ cite journal |author=Bhaskar Bagchi |title=A Joint Universality Theorem for Dirichlet L-Functions
 |journal=Mathematische Zeitschrift |issn=0025-5874 |volume=181 |issue=3
 |date=1982 |pages=319–334 |doi=10.1007/bf01161980|s2cid=120930513
 }}</ref><ref>{{cite book |last=Steuding |first=Jörn |date=2007 |title=Value-Distribution of L-Functions
 |volume=1877 |location=Berlin |publisher=Springer |page=19 |isbn=978-3-540-26526-9 |series=Lecture Notes in Mathematics
 |doi=10.1007/978-3-540-44822-8|arxiv=1711.06671 }}</ref>

===Estimates of the maximum of the modulus of the zeta function===
Let the functions {{math|''F''(''T'';''H'')}} and {{math|''G''(''s''<sub>0</sub>;Δ)}} be defined by the equalities

: <math> F(T;H) = \max_{|t-T|\le H}\left|\zeta\left(\tfrac{1}{2}+it\right)\right|,\qquad G(s_{0};\Delta) = \max_{|s-s_{0}|\le\Delta}|\zeta(s)|. </math>

Here {{mvar|T}} is a sufficiently large positive number, {{math|0 < ''H'' ≪ log log ''T''}}, {{math|''s''<sub>0</sub> {{=}} ''σ''<sub>0</sub> + ''iT''}}, {{math|{{sfrac|1|2}} ≤ ''σ''<sub>0</sub> ≤ 1}}, {{math|0 < Δ < {{sfrac|1|3}}}}. Estimating the values {{mvar|F}} and {{mvar|G}} from below shows, how large (in modulus) values {{math|''ζ''(''s'')}} can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip {{math|0 ≤ Re(''s'') ≤ 1}}.

The case {{math|''H'' ≫ log log ''T''}} was studied by [[Kanakanahalli Ramachandra]]; the case {{math|Δ > ''c''}}, where {{math|''c''}} is a sufficiently large constant, is trivial.

[[Anatolii Alexeevitch Karatsuba|Anatolii Karatsuba]] proved,<ref>{{cite journal| first=A. A.| last=Karatsuba| title= Lower bounds for the maximum modulus of {{math|''ζ''(''s'')}} in small domains of the critical strip | pages=796–798| journal= Mat. Zametki| volume=70|issue=5| year=2001}}</ref><ref>{{cite journal| first=A. A.| last=Karatsuba| title= Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line| pages=99–104| journal= Izv. Ross. Akad. Nauk, Ser. Mat.| volume=68|issue=8| year=2004| doi=10.1070/IM2004v068n06ABEH000513| bibcode=2004IzMat..68.1157K| s2cid=250796539}}</ref> in particular, that if the values {{mvar|H}} and {{math|Δ}} exceed certain sufficiently small constants, then the estimates

: <math> F(T;H) \ge T^{- c_1},\qquad G(s_0; \Delta) \ge T^{-c_2}, </math>

hold, where {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are certain absolute constants.

===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>