Editing Riemann zeta function (section)

==Representations==
===Dirichlet series===
An extension of the area of convergence can be obtained by rearranging the original series.<ref name="Knopp">{{cite book|first=Konrad|last=Knopp|title=Theory of Functions, Part Two|url=https://archive.org/details/in.ernet.dli.2015.212186|date=1947|pages=[https://archive.org/details/in.ernet.dli.2015.212186/page/n57/mode/2up 51–55]|publisher=New York, Dover publications}}</ref> The series
:<math>\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right)</math>
converges for {{math|Re(''s'') > 0}}, while
:<math>\zeta(s) =\frac{1}{s-1}\sum_{n=1}^\infty\frac{n(n+1)}{2}\left(\frac{2n+3+s}{(n+1)^{s+2}}-\frac{2n-1-s}{n^{s+2}}\right)</math>
converge even for {{math|Re(''s'') > −1}}. In this way, the area of convergence can be extended to {{math|Re(''s'') > −''k''}} for any negative integer {{math|−''k''}}.

The recurrence connection is clearly visible from the expression valid for {{math|Re(''s'') > −2}} enabling further expansion by integration by parts.

:<math>\begin{aligned}
\zeta(s)= & 1+\frac{1}{s-1}-\frac{s}{2 !}[\zeta(s+1)-1] \\
- & \frac{s(s+1)}{3 !}[\zeta(s+2)-1] \\
& -\frac{s(s+1)(s+2)}{3 !} \sum_{n=1}^{\infty} \int_0^1 \frac{t^3 d t}{(n+t)^{s+3}}
\end{aligned}</math>

===Mellin-type integrals===
The [[Mellin transform]] of a function {{math|''f''(''x'')}} is defined as<ref>{{cite journal |last=Riemann| first=Bernhard |title=[[On the number of primes less than a given magnitude]]|year=1859|journal=Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin}} translated and reprinted in {{cite book|last=Edwards|first=H. M. |authorlink=Harold Edwards (mathematician) |year=1974 |title=Riemann's Zeta Function |publisher=Academic Press |location=New York |isbn=0-12-232750-0 |zbl=0315.10035}}</ref>

:<math> \int_0^\infty f(x)x^s\, \frac{\mathrm{d}x}{x} </math>

in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of {{mvar|s}} is greater than one, we have

:<math>\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} \,\mathrm{d}x \quad</math> and <math>\quad\Gamma(s)\zeta(s) =\frac1{2s}\int_0^\infty\frac{x^{s}}{\cosh(x)-1} \,\mathrm{d}x</math>,

where {{math|Γ}} denotes the [[gamma function]]. By modifying the [[Contour integration|contour]], Riemann showed that

:<math>2\sin(\pi s)\Gamma(s)\zeta(s) =i\oint_H \frac{(-x)^{s-1}}{e^x-1}\,\mathrm{d}x </math>

for all {{mvar|s}}<ref>Trivial exceptions of values of {{mvar|s}} that cause removable singularities are not taken into account throughout this article.</ref> (where {{mvar|H}} denotes the [[Hankel contour]]).

We can also find expressions which relate to prime numbers and the [[prime number theorem]]. If {{math|''π''(''x'')}} is the [[prime-counting function]], then

:<math>\ln \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x,</math>

for values with {{math|Re(''s'') > 1}}.

A similar Mellin transform involves the Riemann function {{math|''J''(''x'')}}, which counts prime powers {{math|''p''<sup>''n''</sup>}} with a weight of {{math|{{sfrac|1|''n''}}}}, so that

: <math>J(x) = \sum \frac{\pi\left(x^\frac{1}{n}\right)}{n}.</math>

Now

:<math>\ln \zeta(s) = s\int_0^\infty J(x)x^{-s-1}\,\mathrm{d}x. </math>

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's [[prime-counting function]] is easier to work with, and {{math|''π''(''x'')}} can be recovered from it by [[Möbius inversion formula|Möbius inversion]].

===Theta functions===
The Riemann zeta function can be given by a Mellin transform<ref>{{Cite book |first=Jürgen |last=Neukirch |title=Algebraic number theory |publisher=Springer |date=1999 |page=422 |isbn=3-540-65399-6}}</ref>

:<math>2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \int_0^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t,</math>

in terms of [[Theta function|Jacobi's theta function]]

:<math>\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}.</math>

However, this integral only converges if the real part of {{mvar|s}} is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all {{mvar|s}} except 0 and 1:

:<math> \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(it)-t^{-\frac12}\right)t^{\frac{s}{2}-1}\,\mathrm{d}t + \frac{1}{2}\int_1^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t.</math>

===Laurent series===
The Riemann zeta function is [[meromorphic]] with a single [[pole (complex analysis)|pole]] of order one at {{math|''s'' {{=}} 1}}. It can therefore be expanded as a [[Laurent series]] about {{math|''s'' {{=}} 1}}; the series development is then<ref>{{cite journal | last1 = Hashimoto | first1 = Yasufumi | last2 = Iijima | first2 = Yasuyuki | last3 = Kurokawa | first3 = Nobushige | last4 = Wakayama | first4 = Masato | doi = 10.36045/bbms/1102689119 | issue = 4 | journal = [[Simon Stevin (journal)|Bulletin of the Belgian Mathematical Society, Simon Stevin]] | mr = 2115723 | pages = 493–516 | title = Euler's constants for the Selberg and the Dedekind zeta functions | url = https://projecteuclid.org/euclid.bbms/1102689119 | volume = 11 | year = 2004| doi-access = free }}</ref>

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

The constants {{math|''γ''<sub>''n''</sub>}} here are called the [[Stieltjes constants]] and can be defined by the [[limit of a sequence|limit]]

: <math> \gamma_n = \lim_{m \rightarrow \infty}{\left(\left(\sum_{k = 1}^m \frac{(\ln k)^n}{k}\right) - \frac{(\ln m)^{n+1}}{n+1}\right)}.</math>

The constant term {{math|''γ''<sub>0</sub>}} is the [[Euler–Mascheroni constant]].

=== Integral ===
For all {{math|''s'' ∈ ℂ}}, {{math|''s'' ≠ 1}}, the integral relation (cf. [[Abel–Plana formula]])
:<math>\ \zeta(s)\ =\ \frac{ 1 }{\ s - 1\ } + \frac{\ 1\ }{ 2 } + 2 \int_0^{\infty} \frac{ \sin(\ s\ \arctan t\ ) }{\ \left( 1 + t^2 \right)^{s/2} \left( e^{2\pi t} - 1 \right)\ }\ \operatorname{d} t\ </math>
holds true, which may be used for a numerical evaluation of the zeta function.

===Rising factorial===
Another series development using the [[Pochhammer symbol|rising factorial]] valid for the entire complex plane is <ref name="Knopp"/>

:<math>\zeta(s) = \frac{s}{s-1} - \sum_{n=1}^\infty \bigl(\zeta(s+n)-1\bigr)\frac{s(s+1)\cdots(s+n-1)}{(n+1)!}.</math>

This can be used recursively to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the [[Gauss–Kuzmin–Wirsing operator]] acting on {{math|''x''<sup>''s'' − 1</sup>}}; that context gives rise to a series expansion in terms of the [[falling factorial]].<ref>{{cite web|url=http://linas.org/math/poch-zeta.pdf |title=A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator |website=Linas.org |access-date=2017-01-04}}</ref>

===Hadamard product===
On the basis of [[Weierstrass factorization theorem|Weierstrass's factorization theorem]], [[Hadamard]] gave the [[infinite product]] expansion

:<math>\zeta(s) = \frac{e^{\left(\log(2\pi)-1-\frac{\gamma}{2}\right)s}}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^\frac{s}{\rho},</math>

where the product is over the non-trivial zeros {{mvar|ρ}} of {{math|''ζ''}} and the letter {{mvar|γ}} again denotes the [[Euler–Mascheroni constant]]. A simpler [[infinite product]] expansion is

:<math>\zeta(s) = \pi^\frac{s}{2} \frac{\prod_\rho \left(1 - \frac{s}{\rho} \right)}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)}.</math>

This form clearly displays the simple pole at {{math|''s'' {{=}} 1}}, the trivial zeros at −2,&nbsp;−4,&nbsp;... due to the gamma function term in the denominator, and the non-trivial zeros at {{math|''s'' {{=}} ''ρ''}}. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form {{mvar|ρ}} and {{math|1 − ''ρ''}} should be combined.)

===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>

===Rapidly convergent series===
[[Peter Borwein]] developed an algorithm that applies [[Chebyshev polynomial]]s to the [[Dirichlet eta function]] to produce a [[Dirichlet eta function#Borwein's method|very rapidly convergent series suitable for high precision numerical calculations]].<ref>{{cite book|first = Peter|last = Borwein|author-link = Peter Borwein|chapter-url = http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf|chapter = An Efficient Algorithm for the Riemann Zeta Function|series = Conference Proceedings, Canadian Mathematical Society|year = 2000|title = Constructive, Experimental, and Nonlinear Analysis|volume = 27|pages = 29–34|isbn = 978-0-8218-2167-1|editor-first = Michel A.|editor-last = Théra|publisher = [[American Mathematical Society]], on behalf of the [[Canadian Mathematical Society]]|location = Providence, RI|access-date = 25 November 2017|archive-date = 26 July 2011|archive-url = https://web.archive.org/web/20110726090927/http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf|url-status = dead}}</ref>

===Series representation at positive integers via the primorial===
: <math> \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)}\qquad k=2,3,\ldots.</math>
Here {{math|''p<sub>n</sub>''#}} is the [[primorial]] sequence and {{math|''J<sub>k</sub>''}} is [[Jordan's totient function]].<ref>{{cite journal
|first=István
|last=Mező
|title=The primorial and the Riemann zeta function
|journal= The American Mathematical Monthly
|year=2013
|volume=120
|issue=4
|page=321
}}</ref>

===Series representation by the incomplete poly-Bernoulli numbers===
The function {{mvar|ζ}} can be represented, for {{math|Re(''s'') > 1}}, by the infinite series
:<math>\zeta(s)=\sum_{n=0}^\infty B_{n,\ge2}^{(s)}\frac{(W_k(-1))^n}{n!},</math>
where {{math|''k'' ∈ {−1, 0{{)}}}}, {{math|''W<sub>k</sub>''}} is the {{mvar|k}}th branch of the [[Lambert W function|Lambert {{mvar|W}}-function]], and {{math|''B''{{su|b=''n'', ≥2|p=(''μ'')}}}} is an incomplete poly-Bernoulli number.<ref>{{cite journal
|first1=Takao
|last1=Komatsu
|first2=István
|last2=Mező
|title=Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers
|journal=Publicationes Mathematicae Debrecen
|year=2016
|volume=88
|issue=3–4
|pages=357–368
|doi=10.5486/pmd.2016.7361
|arxiv=1510.05799
|s2cid=55741906
}}</ref>

===The Mellin transform of the Engel map===
The function <math>g(x) = x \left( 1+\left\lfloor x^{-1}\right\rfloor \right) -1</math> is iterated to find the coefficients appearing in [[Engel expansion]]s.<ref>{{Cite web|url=http://oeis.org/A220335|title=A220335 - OEIS|website=oeis.org|access-date=2019-04-17}}</ref>

The [[Mellin transform]] of the map <math>g(x)</math> is related to the Riemann zeta function by the formula
:<math>  \begin{align}
    \int_0^1 g (x) x^{s - 1} \, dx & = \sum_{n = 1}^\infty
    \int_{\frac{1}{n + 1}}^{\frac{1}{n}} (x (n + 1) - 1) x^{s - 1} \, d x\\[6pt]
    & = \sum_{n = 1}^\infty \frac{n^{- s} (s - 1) + (n + 1)^{- s - 1} (n^2 + 2 n + 1) + n^{- s - 1} s - n^{1 - s}}{(s + 1) s (n + 1)}\\[6pt]
    & = \frac{\zeta (s + 1)}{s + 1} - \frac{1}{s (s + 1)}
  \end{align}</math>

===Thue-Morse sequence===
Certain linear combinations of Dirichlet series whose coefficients are terms of the [[Thue-Morse sequence]] give rise to identities involving the Riemann Zeta function.<ref>
{{cite journal|author1-link=Tóth|last1=Tóth|first1=László|title=Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence|journal=Integers|volume=22|year=2022|issue=article 98|arxiv=2211.13570 }}
</ref> For instance:
:<math>  \begin{align}
    \sum_{n\geq1} \frac{5 t_{n-1} + 3 t_n}{n^2} &= 4 \zeta(2) = \frac{2 \pi^2}{3}, \\
	\sum_{n\geq1} \frac{9 t_{n-1} + 7 t_n}{n^3} &= 8 \zeta(3),\end{align}</math>
where <math>(t_n)_{n\geq0}</math> is the <math>n^{\rm th}</math> term of the Thue-Morse sequence. In fact, for all <math>s</math> with real part greater than <math>1</math>, we have
:<math>  (2^s+1) \sum_{n\geq1} \frac{t_{n-1}}{n^s} + (2^s-1) \sum_{n\geq1} \frac{t_{n}}{n^s} = 2^s \zeta(s).</math>