Editing Riemann zeta function (section)

===Globally convergent series===
A globally convergent series for the zeta function, valid for all complex numbers {{mvar|s}} except {{math|''s'' {{=}} 1 + {{sfrac|2π''i''|ln 2}}''n''}} for some integer {{mvar|n}}, was conjectured by [[Konrad Knopp]] in 1926 <ref name="blag2018" /> and proven by [[Helmut Hasse]] in 1930<ref name = Hasse1930 /> (cf. [[Euler summation]]):

:<math>\zeta(s)=\frac{1}{1-2^{1-s}} \sum_{n=0}^\infty \frac {1}{2^{n+1}} \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{(k+1)^{s}}.</math>

The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.<ref>{{cite journal|first = Jonathan|last = Sondow|title = Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series|journal = [[Proceedings of the American Mathematical Society]]|year = 1994|volume = 120|issue = 2|pages = 421–424|url = https://www.ams.org/journals/proc/1994-120-02/S0002-9939-1994-1172954-7/S0002-9939-1994-1172954-7.pdf|doi = 10.1090/S0002-9939-1994-1172954-7|doi-access = free}}</ref>

Hasse also proved the globally converging series
:<math>\zeta(s)=\frac 1{s-1}\sum_{n=0}^\infty \frac 1{n+1}\sum_{k=0}^n\binom {n}{k}\frac{(-1)^k}{(k+1)^{s-1}}</math>
in the same publication.<ref name = Hasse1930 /> Research by Iaroslav Blagouchine<ref>{{cite journal
 | last = Blagouchine | first = Iaroslav V.
 | arxiv = 1501.00740
 | doi = 10.1016/j.jnt.2015.06.012
 | journal = [[Journal of Number Theory]]
 | pages = 365–396
 | title = Expansions of generalized Euler's constants into the series of polynomials in {{pi}}<sup>&minus;2</sup> and into the formal enveloping series with rational coefficients only
 | volume = 158
 | year = 2016}}</ref><ref name="blag2018">{{cite journal
 | last = Blagouchine | first = Iaroslav V.
 | arxiv = 1606.02044
 | url = http://math.colgate.edu/~integers/vol18a.html
 | journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
 | pages = 1–45
 | title = Three Notes on Ser's and Hasse's Representations for the Zeta-functions
 | volume = 18A
 | year = 2018| doi = 10.5281/zenodo.10581385
 | bibcode = 2016arXiv160602044B}}</ref>
has found that a similar, equivalent series was published by [[Joseph Ser]] in 1926.<ref>{{cite journal|first = Joseph|last = Ser|author-link = Joseph Ser|title = Sur une expression de la fonction ζ(s) de Riemann|trans-title = Upon an expression for Riemann's ζ function|year = 1926|journal = [[Comptes rendus hebdomadaires des séances de l'Académie des Sciences]]|volume = 182|pages = 1075–1077|language = fr}}</ref>

In 1997 K. Maślanka gave another globally convergent (except {{math|s {{=}} 1}}) series for the Riemann zeta function:

:<math>\zeta (s)=\frac{1}{s-1}\sum_{k=0}^\infty \biggl(\prod_{i=1}^{k} (i-\frac{s}{2})\biggl) \frac{A_{k}}{k!}=
\frac{1}{s-1} \sum_{k=0}^\infty \biggl(1-\frac{s}{2}\biggl)_{k}
\frac{A_{k}}{k!}</math>

where real coefficients <math>A_k</math> are given by:

:<math>A_k=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}(2j+1)\zeta
(2j+2)=\sum_{j=0}^{k}\binom{k}{j}\frac{B_{2j+2}\pi ^{2j+2}}{\left(2\right) _{j}\left( \frac{1}{2}\right) _{j}} </math>

Here <math>B_{n}</math> are the Bernoulli numbers and <math>(x)_{k}</math> denotes the Pochhammer symbol.<ref>{{cite journal
 |first = Krzysztof
 |last = Maślanka
 |title = The Beauty of Nothingness
 |year = 1997
 |journal = Acta Cosmologica
 |volume = XXIII-I
 |pages = 13–17}}</ref><ref>{{cite journal
 |first = Luis
 |last = Báez-Duarte
 |title = On Maslanka's Representation for the Riemann Zeta Function
 |year = 2010
 |journal = [[International Journal of Mathematics and Mathematical Sciences]]
 |volume = 2010
 |pages = 1–9
|doi = 10.1155/2010/714147
 |doi-access = free
 |arxiv = math/0307214
 }}</ref>

Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points <math>s=2,4,6,\ldots </math>, i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on [[Carlson's theorem]], was presented by Philippe Flajolet in 2006.<ref>{{cite journal
 |first1 = Philippe
 |last1 = Flajolet
 |first2 = Linas
 |last2 = Vepstas
 |title = On Differences of Zeta Values
 |year = 2008
 |journal = [[Journal of Computational and Applied Mathematics]]
 |volume = 220
 |issue = 1–2 October
 |pages = 58–73
 |doi = 10.1016/j.cam.2007.07.040
 |arxiv = math/0611332|bibcode = 2008JCoAM.220...58F
 }}</ref>

The asymptotic behavior of the coefficients <math>A_{k}</math> is rather curious: for growing <math>k</math> values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as <math>k^{-2/3}</math>). Using the saddle point method, we can show that

:<math>A_{k}\sim \frac{4\pi ^{3/2}}{\sqrt{3\kappa }}\exp \biggl( -\frac{3\kappa }{2}+\frac{\pi ^{2}}{4\kappa }\biggl) \cos \biggl( \frac{4\pi }{3}-\frac{3\sqrt{3}
\kappa }{2}+\frac{\sqrt{3}\pi ^{2}}{4\kappa }\biggl)</math>

where <math>\kappa</math> stands for:

:<math>\kappa :=\sqrt[3]{\pi ^{2}k} </math>

(see <ref>{{cite journal
 |first1 = Krzysztof
 |last1 = Maślanka
 |first2 = Andrzej
 |last2 = Koleżyński
 |title = The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm
 |journal = Computational Methods in Science and Technology
 |year = 2022
 |volume = 28
 |issue = 2
 |pages = 47–59
|doi = 10.12921/cmst.2022.0000014
 |arxiv = 2210.04609
 |s2cid = 252780397
 }}</ref> for details).

On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.<ref>{{cite journal
 |first = Luis
 |last = Báez-Duarte
 |title = A New Necessary and Sufficient Condition for the Riemann Hypothesis
 |journal = Number Theory
 |arxiv = math/0307215
 |year = 2003
|bibcode = 2003math......7215B
 }}</ref><ref>{{cite journal
 |first = Krzysztof
 |last = Maślanka
 |title = Báez-Duarte's Criterion for the Riemann Hypothesis and Rice's Integrals
 |journal = Number Theory
 |arxiv = math/0603713v2
 |year = 2006
|bibcode = 2006math......3713M
 }}</ref><ref>{{cite journal
 |first = Marek
 |last = Wolf
 |title = Some remarks on the Báez-Duarte criterion for the Riemann Hypothesis
 |journal = Computational Methods in Science and Technology
 |volume = 20
 |year = 2014
 |issue = 2
 |pages = 39–47
|doi = 10.12921/cmst.2014.20.02.39-47
 |doi-access = free
 }}</ref> Namely, if we define the coefficients <math>c_{k}</math> as

:<math>c_{k}:=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\frac{1}{\zeta (2j+2)}</math>

then the Riemann hypothesis is equivalent to

:<math>c_{k}=\mathcal{O}\biggl( k^{-3/4+\varepsilon }\biggl) \qquad (\forall\varepsilon >0) </math>