View source for Riemann zeta function

{{infobox mathematics function
| domain = <math>\Complex\setminus\{1\}</math>
| codomain = <math>\Complex</math>
| image = File:Riemann-Zeta-Func.png
| caption = The Riemann zeta function {{math|''ζ''(''z'')}} plotted with [[domain coloring]].<ref>{{cite web|url=http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb |title=Jupyter Notebook Viewer|website=Nbviewer.ipython.org |date= |accessdate=2017-01-04}}</ref>
| zero = <math>-\frac{1}{2}</math>
| plusinf = <math>1</math>
| vr1 = <math>2</math>
| f1 = <math>\frac{\pi^2}{6}</math>
| vr2 = <math>-1</math>
| f2 = <math>-{1\over12}</math>
| vr3 = <math>-2</math>
| f3 = <math>0</math>
}}
[[File:Riemann-Zeta-Detail.png|right|thumb|200px|The pole at <math>z=1</math>, and two zeros on the critical line.]]

The '''Riemann zeta function''' or '''Euler–Riemann zeta function''', {{math|''ζ''(''s'')}}, is a [[function (mathematics)|function]] of a [[complex variable]] ''s'' that [[analytic continuation|analytically continues]] the sum of the [[Dirichlet series]] 
:<math>\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s}</math>

which converges when the [[real part]] of {{mvar|s}} is greater than&nbsp;1. More general [[#Representations|representations]] of {{math|''ζ''(''s'')}} for all {{mvar|s}} are given below. The Riemann zeta function plays a pivotal role in [[analytic number theory]] and has applications in [[physics]], [[probability theory]], and applied [[statistics]].

As a function of a real variable, [[Leonhard Euler]] first introduced and studied it in the first half of the eighteenth century without using [[complex analysis]], which was not available at the time.  [[Bernhard Riemann]]'s 1859 article "[[On the Number of Primes Less Than a Given Magnitude]]"  extended the Euler definition to a [[complex number|complex]] variable, proved its [[meromorphic]] continuation and [[functional equation]], and established a relation between its [[Root of a function|zeros]] and [[prime number theorem|the distribution of prime numbers]].<ref>This paper also contained the [[Riemann hypothesis]], a [[conjecture]] about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be the most important unsolved problem in [[pure mathematics]].{{cite web|last=Bombieri|first= Enrico|url=http://www.claymath.org/sites/default/files/official_problem_description.pdf|title=The Riemann Hypothesis – official problem description|publisher=[[Clay Mathematics Institute]]|accessdate=2014-08-08}}</ref>

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, {{math|''ζ''(2)}}, provides a solution to the [[Basel problem]]. In 1979 [[Roger Apéry]] proved the irrationality of {{math|[[Apéry's constant|''ζ''(3)]]}}. The values at negative integer points, also found by Euler, are [[rational number]]s and play an important role in the theory of [[modular form]]s. Many generalizations of the Riemann zeta function, such as [[Dirichlet series]], [[Dirichlet L-function|Dirichlet {{mvar|L}}-functions]] and [[L-function|{{mvar|L}}-functions]], are known.

==Definition==
[[File:Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.pdf|thumb|Bernhard Riemann's article ''On the number of primes below a given magnitude''.]]

The Riemann zeta function {{math|''ζ''(''s'')}} is a function of a complex variable {{math|''s'' {{=}} ''σ'' + ''it''}}.  (The notation {{mvar|s}}, {{mvar|σ}}, and {{mvar|t}} is used traditionally in the study of the zeta function, following Riemann.)

The zeta function can be expressed by the following integral:

<!-- This seemingly roundabout way of writing the integral makes it clear that the zeta function is a quotient of two Mellin transforms
 i.e. that we integrate 1/(e^x -1) against the invariant measure of R^* and the multiplicative character character  x^s -->
:<math> \zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x </math>

where 
:<math>\Gamma(s) = \int_0^\infty x^{s-1}\,e^{-x} \, \mathrm{d}x </math>

is the [[gamma function]].

For the special case where the real part of {{mvar|s}} is greater than&nbsp;1, {{math|''ζ''(''s'')}} always converges, and can be simplified to the following [[infinite series]]:

:<math>\zeta(s) = \sum_{n=1}^\infty n^{-s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \quad \sigma = \operatorname{Re}(s) > 1.</math>

The Riemann zeta function is defined as the [[analytic continuation]] of the function defined for {{math|''σ'' > 1}} by the sum of the preceding series.

[[Leonhard Euler]] considered the above series in 1740 for positive integer values of {{mvar|s}}, and later [[Chebyshev]] extended the definition to {{math|Re(''s'') > 1}}.<ref name='devlin'>{{cite book | last = Devlin | first = Keith | authorlink = Keith Devlin | title = The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time | publisher = Barnes & Noble | year = 2002 | location = New York | pages = 43–47 | isbn = 978-0-7607-8659-8}}</ref>

The above series is a prototypical [[Dirichlet series]] that [[absolute convergence|converges absolutely]] to an [[analytic function]] for {{mvar|s}} such that {{math|''σ'' > 1}} and [[divergent series|diverges]] for all other values of {{mvar|s}}. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values {{math|''s'' ≠ 1}}. For {{math|''s'' {{=}} 1}} the series is the [[harmonic series (mathematics)|harmonic series]] which diverges to {{math|[[Extended real number line|+∞]]}}, and

: <math> \lim_{s\to 1}(s-1)\zeta(s)=1.</math>

Thus the Riemann zeta function is a [[meromorphic function]] on the whole complex {{mvar|s}}-plane, which is [[holomorphic function|holomorphic]] everywhere except for a [[simple pole]] at {{math|''s'' {{=}} 1}} with [[Residue (complex analysis)|residue]] 1.

==Specific values==
{{main|Particular values of the Riemann zeta function}}
For any positive even integer {{math|2''n''}}:

:<math> \zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}</math>

where {{math|''B''<sub>2''n''</sub>}} is the {{math|2''n''}}th [[Bernoulli number]].

For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic {{mvar|K}}-theory of the integers; see [[Special values of L-functions|Special values of {{mvar|L}}-functions]].

For nonpositive integers, one has

:<math>\zeta(-n)=(-1)^n \frac{B_{n+1}}{n+1}</math>

for  {{math|''n'' ≥ 0}} (using the NIST convention that {{math|''B''<sub>1</sub> {{=}} −{{sfrac|1|2}}}})

In particular, {{mvar|ζ}} vanishes at the negative even integers because {{math|''B''<sub>''m''</sub> {{=}} 0}} for all odd {{mvar|m}} other than&nbsp;1. These are the so-called "trivial zeros" of the zeta function.

Via [[analytic continuation]], one can show that: 
*<math>\zeta(-1) = -\tfrac{1}{12}</math>
::This gives a way to assign a finite result to the divergent series [[1 + 2 + 3 + 4 + ⋯]], which has been used in certain contexts such as [[string theory]].<ref name='polchinski'>{{cite book | last = Polchinski | first = Joseph | authorlink = Joseph Polchinski | title = String Theory, Volume I: An Introduction to the Bosonic String | publisher = Cambridge University Press | year = 1998 | page = 22 | isbn = 978-0-521-63303-1}}</ref>
*<math>\zeta(0) = -\tfrac{1}{2};</math>
::Similarly to the above, this assigns a finite result to the series [[1 + 1 + 1 + 1 + ⋯]].
*<math>\zeta\bigl(\tfrac12\bigr) \approx -1.46035 45088 09586 81289</math> &nbsp; ({{OEIS2C|id=A059750}})
:: This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.<ref>{{cite journal|first=A. J. |last=Kainz |first2=U. M. |last2=Titulaer |title=An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations |pages=1855–1874 |journal=J. Phys. A: Math. Gen. |volume=25 |issue=7 |date=1992|bibcode=1992JPhA...25.1855K |doi=10.1088/0305-4470/25/7/026 }}</ref>

*<math>\zeta(1) = 1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots = \infty;</math>
::If we approach from numbers larger than 1, this is the [[harmonic series (mathematics)|harmonic series]]. But its [[Cauchy principal value]]
:::<math> \lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}</math>
::exists which is the [[Euler–Mascheroni constant]] {{math|''γ'' {{=}} 0.5772…}}.

*<math>\zeta\bigl(\tfrac32\bigr) \approx 2.61237 53486 85488 34335;</math> &nbsp; ({{OEIS2C|id=A078434}})
:: This is employed in calculating the critical temperature for a [[Bose–Einstein condensate]] in a box with periodic boundary conditions, and for [[spin wave]] physics in magnetic systems.

*<math>\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.64493 40668 48226 43647;\!</math> &nbsp; ({{OEIS2C|id=A013661}})
:: The demonstration of this equality is known as the [[Basel problem]]. The reciprocal of this sum answers the question: ''What is the probability that two numbers selected at random are [[coprime|relatively prime]]?''<ref>{{cite book|authorlink=C. Stanley Ogilvy|first=C. S. |last=Ogilvy |first2=J. T. |last2=Anderson |title=Excursions in Number Theory |pages=29–35 |publisher=Dover Publications |date=1988 |isbn=0-486-25778-9}}</ref>

*<math>\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.20205 69031 59594 28540;</math> &nbsp; ({{OEIS2C|id=A002117}})

:: This number is called [[Apéry's constant]].

*<math>\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.08232 32337 11138 19152;</math> &nbsp; ({{OEIS2C|id=A0013662}})

:: This appears when integrating [[Planck's law]] to derive the [[Stefan–Boltzmann law]] in physics.

==Euler product formula==
The connection between the zeta function and [[prime number]]s was discovered by Euler, who [[Proof of the Euler product formula for the Riemann zeta function|proved the identity]]

:<math>\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}},</math>

where, by definition, the left hand side is {{math|''ζ''(''s'')}} and the [[infinite product]] on the right hand side extends over all prime numbers {{mvar|p}} (such expressions are called [[Euler product]]s):

:<math>\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}} \cdots \frac{1}{1-p^{-s}} \cdots</math>

Both sides of the Euler product formula converge for {{math|Re(''s'') > 1}}. The [[Proof of the Euler product formula for the Riemann zeta function|proof of Euler's identity]] uses only the formula for the [[geometric series]] and the [[fundamental theorem of arithmetic]]. Since the [[harmonic series (mathematics)|harmonic series]], obtained when {{math|''s'' {{=}} 1}}, diverges, Euler's formula (which becomes {{math|∏<sub>''p''</sub> {{sfrac|''p''|''p'' − 1}}}}) implies that there are [[Euclid's theorem|infinitely many primes]].<ref>{{cite book|first=Charles Edward |last=Sandifer |title=How Euler Did It |publisher=Mathematical Association of America |date=2007 |page=193 |isbn=978-0-88385-563-8}}</ref>

The Euler product formula can be used to calculate the [[asymptotic density|asymptotic probability]] that {{mvar|s}} randomly selected integers are set-wise [[coprime]]. Intuitively, the probability that any single number is divisible by a prime (or any integer) {{mvar|p}} is {{math|{{sfrac|1|''p''}}}}. Hence the probability that {{mvar|s}} numbers are all divisible by this prime is {{math|{{sfrac|1|''p''{{isup|''s''}}}}}}, and the probability that at least one of them is ''not'' is {{math|1 − {{sfrac|1|''p''{{isup|''s''}}}}}}. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors {{mvar|n}} and {{mvar|m}} if and only if it is divisible by&nbsp;{{mvar|nm}}, an event which occurs with probability&nbsp;{{math|{{sfrac|1|''nm''}}}}). Thus the asymptotic probability that {{mvar|s}} numbers are coprime is given by a product over all primes,

: <math>\prod_{p \text{ prime}} \left(1-\frac{1}{p^s}\right) = \left( \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \right)^{-1} = \frac{1}{\zeta(s)}. </math>

(More work is required to derive this result formally.)<ref>{{cite journal|first=J. E. |last=Nymann|title=On the probability that {{mvar|k}} positive integers are relatively prime|journal=Journal of Number Theory|volume=4|year=1972|pages=469–473|doi=10.1016/0022-314X(72)90038-8|issue=5|bibcode = 1972JNT.....4..469N }}</ref>

== Riemann's functional equation ==
The zeta function satisfies the [[functional equation]]:
:<math>\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s),</math>

where {{math|Γ(''s'')}} is the [[gamma function]]. This is an equality of meromorphic functions valid on the whole [[complex plane]]. The equation relates values of the Riemann zeta function at the points {{mvar|s}} and {{math|1 − ''s''}}, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that {{math|''ζ''(''s'')}} has a simple zero at each even negative integer {{math|''s'' {{=}} −2''n''}}, known as the '''[[Triviality (mathematics)|trivial]] zeros''' of {{math|''ζ''(''s'')}}. When {{mvar|s}} is an even positive integer, the product {{math|sin({{sfrac|{{pi}}''s''|2}})Γ(1 − ''s'')}} on the right is non-zero because {{math|Γ(1 − ''s'')}} has a simple [[pole (complex analysis)|pole]], which cancels the simple zero of the sine factor.

{{Collapse top|title=Proof of functional equation}}
A proof of the functional equation proceeds as follows:
We observe that if <math>\sigma > 0</math>, then

: <math>\int_0^\infty x^{{1\over2}{s} - 1}e^{-n^2\pi x}\, dx = {\Gamma \left({s\over2}\right)\over{n^s\pi^{s\over2}}}.</math>

As a result, if <math>\sigma > 1</math> then

: <math> \frac{\Gamma\left(\frac s 2\right)\zeta(s)}{\pi^{s/2}} = \sum_{n=1}^\infty \int\limits_0^\infty x^{{s\over 2}-1} e^{-n^2 \pi x}\, dx = \int_0^\infty x^{{s\over 2}-1} \sum_{n=1}^\infty e^{-n^2 \pi x}\, dx.</math>

With the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on <math>\sigma</math>)

For convenience, let

: <math>\psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}</math>

Then <math>\zeta(s) = {\pi^{s\over2}\over\Gamma({s \over 2})} \int\limits_0^\infty x^{{1\over2}{s} - 1}\psi(x)\, dx</math>

Given that <math>\sum_{n=-\infty}^\infty {e^{-n^2 \pi x}} = {1 \over \sqrt x}\sum_{n=-\infty}^\infty {e^{-n^2 \pi \over x}}</math>

Then <math>2 \psi(x)+1 = {1\over \sqrt x} \left\{2\psi \left ( {1 \over x} \right )+1\right\}</math>

Hence <math>\pi^{-{s \over 2}} \Gamma \left ( {s \over 2} \right ) \zeta (s) = \int_0^1 x^{{s\over 2}-1} \psi(x) \, dx + \int_1^\infty x^{{s\over 2}-1} \psi(x) \, dx</math>

This is equivalent to <math>\int\limits_0^1 x^{{s\over 2}-1} \left\{ {1\over \sqrt x} \psi \left ( {1 \over x} \right ) + {1\over 2 \sqrt x} - {1 \over 2} \right\} \, dx + \int\limits_1^\infty x^{{s\over 2}-1} \psi(x) \, dx</math>

Or

: <math>
\begin{align}
& {1 \over {s-1}} - {1\over s} + \int\limits_0^1 x^{{{s}\over 2}-{3\over 2}} \psi \left ( {1 \over x} \right ) \, dx + \int\limits_1^\infty x^{{{s}\over 2}-1} \psi(x) \, dx \\[5pt]
= {} & {1 \over {s({s-1})}} + \int\limits_1^\infty \left ({x^{-{{s}\over 2}-{1\over 2}} + x^{{{s}\over 2}-1}} \right ) \psi(x) \, dx
\end{align}
</math>

which is convergent for all ''s'', so holds by analytic continuation. Furthermore, the RHS is unchanged if ''s'' is changed to 1&nbsp;−&nbsp;''s''. Hence

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

which is the functional equation.
{{cite book|title=The Theory of the Riemann Zeta-function|publisher=Oxford Science Publications|date={{date|1986}}|edition=2nd|pages=21–22|author=E. C. Titchmarsh|location=[[Oxford]]|isbn=0-19-853369-1}} Attributed to [[Bernhard Riemann]].
{{Collapse bottom}}

The functional equation was established by Riemann in his 1859 paper "[[On the Number of Primes Less Than a Given Magnitude]]" and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the [[Dirichlet eta function]] (alternating zeta function):

: <math>\eta(s)= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} = \left(1-{2^{1-s}}\right)\zeta(s).</math>

Incidentally, this relation gives an equation for calculating {{math|''ζ''(''s'')}} in the region  0 < {{math|Re(''s'')}} < 1, i.e.

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

where the ''&eta;''-series is [[Convergent series|convergent]] (albeit [[absolute convergence|non-absolutely]]) in the larger half-plane {{math|''s'' > 0}} (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine<ref>[http://www.mathnet.ru/php/seminars.phtml?&presentid=19339&option_lang=eng I. V. Blagouchine ''The history of the functional equation of the zeta-function.'' Seminar on the History of Mathematics, Steklov Institute of Mathematics at St. Petersburg, 1 March 2018.] [https://iblagouchine.perso.centrale-marseille.fr/Blagouchine-The-history-of-the-functional-equation-of-the-zeta-function-(1-March-2018).php PDF]</ref><ref>[https://link.springer.com/article/10.1007/s11139-013-9528-5 I. V. Blagouchine ''Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results.'' The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Addendum: vol. 42, pp. 777–781, 2017.] [https://iblagouchine.perso.centrale-marseille.fr/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).php PDF]</ref>).

Riemann also found a symmetric version of the functional equation applying to the xi-function:

:<math>\xi(s) = \frac{1}{2}\pi^{-\frac{s}{2}}s(s-1)\Gamma\left(\frac{s}{2}\right)\zeta(s),\!</math>

which satisfies:

:<math>\xi(s) = \xi(1 - s).\!</math>

(Riemann's [[Riemann Ξ function|original {{math|''ξ''(''t'')}}]] was slightly different.)

==Zeros, the critical line, and the Riemann hypothesis==
{{main|Riemann hypothesis}}
[[File:Zero-free region for the Riemann zeta-function.svg|right|thumb|300px|Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of {{math|''σ'' {{=}} 1}} and to the left of {{math|''σ'' {{=}} 0}} (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line {{math|''σ'' {{=}} {{sfrac|1|2}}}} and, according to the [[Riemann hypothesis]], they all lie on the line {{math|''σ'' {{=}} {{sfrac|1|2}}}}.]]
[[Image:Zeta polar.svg|right|thumb|300px|This image shows a plot of the Riemann zeta function along the critical line for real values of {{mvar|t}} running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.]]
[[File:RiemannCriticalLine.svg|thumb|300px|The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(''s'') = 1/2. The first non-trivial zeros can be seen at Im(''s'') = ±14.135, ±21.022 and ±25.011.]]
The functional equation shows that the Riemann zeta function has zeros at {{nowrap|−2, −4,…}}. These are called the '''trivial zeros'''. They are trivial in the sense that their existence is relatively easy to prove, for example, from {{math|sin {{sfrac|π''s''|2}}}} being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {{math|{''s'' ∈ '''ℂ''' : 0 < Re(''s'') < 1<nowiki>}</nowiki>}}, which is called the '''critical strip'''. The [[Riemann hypothesis]], considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero {{mvar|s}} has {{math|Re(''s'') {{=}} {{sfrac|1|2}}}}. In the theory of the Riemann zeta function, the set {{math|{''s'' ∈ '''ℂ''' : Re(''s'') {{=}} {{sfrac|1|2}}<nowiki>}</nowiki>}}&nbsp;is called the '''critical line'''. For the Riemann zeta function on the critical line, see [[Z function|{{mvar|Z}}-function]].

=== The Hardy–Littlewood conjectures ===
In 1914, [[G. H. Hardy|Godfrey Harold Hardy]] proved that {{math|''ζ'' ({{sfrac|1|2}} + ''it'')}} has infinitely many real zeros.

Hardy and [[John Edensor Littlewood]] formulated two conjectures on the density and distance between the zeros of {{math|''ζ'' ({{sfrac|1|2}} + ''it'')}} on intervals of large positive real numbers. In the following, {{math|''N''(''T'')}} is the total number of real zeros and {{math|''N''<sub>0</sub>(''T'')}} the total number of zeros of odd order of the function {{math|''ζ'' ({{sfrac|1|2}} + ''it'')}} lying in the interval {{math|(0, ''T'']}}.
{{numbered list
|For any {{math|''ε'' > 0}}, there exists a {{math|''T''<sub>0</sub>(''ε'') > 0}} such that when
:<math>T \geq T_0(\varepsilon) \quad\text{ and }\quad H=T^{\frac14+\varepsilon},</math>
the interval {{math|(''T'', ''T'' + ''H'']}} contains a zero of odd order.
|For any {{math|''ε'' > 0}}, there exists a {{math|''T''<sub>0</sub>(''ε'') > 0}} and {{math|''c<sub>ε</sub>'' > 0}} such that the inequality
:<math>N_0(T+H)-N_0(T) \geq c_\varepsilon H</math>
holds when
:<math>T \geq T_0(\varepsilon) \quad\text{ and }\quad H=T^{\frac12+\varepsilon}.</math>
}}
These two conjectures opened up new directions in the investigation of the Riemann zeta function.

=== Zero-free region ===
The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. The [[prime number theorem]] is equivalent to the fact that there are no zeros of the zeta function on the {{math|Re(''s'') {{=}} 1}} line.<ref name="Diamond1982">{{cite journal|first=Harold G.|last=Diamond|title=Elementary methods in the study of the distribution of prime numbers|journal=Bulletin of the American Mathematical Society|volume=7|issue=3|year=1982|pages=553–89|mr=670132|doi=10.1090/S0273-0979-1982-15057-1}}</ref> A better result<ref>{{cite journal | last1 = Ford | first1 = K. | year = 2002 | title = Vinogradov's integral and bounds for the Riemann zeta function | url = | journal = Proc. London Math. Soc. | volume = 85 | issue = 3| pages = 565–633 | doi = 10.1112/S0024611502013655 }}</ref> that follows from an effective form of [[Vinogradov's mean-value theorem]] is that {{math|''ζ'' (''σ'' + ''it'') ≠ 0}} whenever {{math|{{abs|''t''}} ≥ 3}} and

:<math>\sigma\ge 1-\frac{1}{57.54(\log{|t|})^\frac23(\log{\log{|t|}})^\frac13}.</math>

The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound [[Riemann hypothesis#Consequences|consequences]] in the theory of numbers.

=== Other results ===
It is known that there are infinitely many zeros on the critical line. [[John Edensor Littlewood|Littlewood]] showed that if the sequence ({{math|''γ<sub>n</sub>''}}) contains the imaginary parts of all zeros in the [[upper half-plane]] in ascending order, then

:<math>\lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_n\right)=0.</math>

The [[critical line theorem]] asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)

In the critical strip, the zero with smallest non-negative imaginary part is {{math|{{sfrac|1|2}} + 14.13472514…''i''}} ({{OEIS2C|A058303}}). The fact that
:<math>\zeta(s)=\overline{\zeta(\overline{s})}</math>
for all complex {{math|''s'' ≠ 1}} implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line {{math|Re(''s'') {{=}} {{sfrac|1|2}}}}.

==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|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 |volume=54 |issue=1 |date=2010 |pages=209–219
 |jstor=43736941 |doi=10.1090/S0025-5718-1975-0384673-1}}</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}}</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'' ≪ ln ln ''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'' ≫ ln ln ''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}}</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 {{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>

==Representations==
===Dirichlet series===
An extension of the area of convergence can be obtained by rearranging the original series.<ref>{{cite book|first=Konrad|last=Knopp|title=Theory of Functions|url=https://archive.org/details/theoryoffunction00knop|url-access=registration|date=1945|pages=[https://archive.org/details/theoryoffunction00knop/page/51 51–55]}}</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>
converges 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''}}.

===Mellin-type integrals===
{{unreferenced section|date=December 2014}}
The [[Mellin transform]] of a function {{math|''f''(''x'')}} is defined as

:<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, </math>

where {{math|Γ}} denotes the [[gamma function]]. By modifying the 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}} (where {{mvar|H}} denotes the [[Hankel contour]]).

Starting with the integral formula <math> \zeta(n)  {\Gamma(n)} =\int_{0}^{\infty} \frac{x ^ {n-1}}{e ^ x - 1} \mathrm{d}x,</math> one can show<ref>{{cite web|url=https://math.stackexchange.com/q/117529 |title=Evaluating the definite integral...|website=math.stackexchange.com}}</ref> by substitution and iterated differentation for natural <math>k\ge 2</math>
 
:<math>  \int_{0}^{\infty} \frac{x ^ {n}e^x}{(e ^ x - 1)^k} \mathrm{d}x =  \frac{n!}{ (k-1)!}\zeta^n\prod_{j=0}^{k-2}\left(1-\frac j\zeta\right)</math> 
using the notation of [[umbral calculus]] where each power <math>\zeta^r</math> is to be replaced by <math>\zeta(r)</math>, so e.g. for <math>k=2</math>  we have <math>  \int_{0}^{\infty} \frac{x ^ {n}e^x}{(e ^ x - 1)^2} \mathrm{d}x = {n!}\zeta(n) ,</math> while for <math>k=4</math> this becomes    
:<math>  \int_{0}^{\infty} \frac{x ^ {n}e^x}{(e ^ x - 1)^4} \mathrm{d}x =  \frac{n!}{ 6} \bigl( \zeta^{n-2} -3\zeta^{n-1} +2\zeta^n \bigr)=  n!\frac{  \zeta(n-2) -3\zeta(n-1) +2\zeta(n)  }{ 6}.</math>

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 prime-counting 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 we have

:<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===
{{unreferenced section|date=December 2014}}
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

:<math>\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n\gamma_n}{n!}(s-1)^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)^\frac{s}{2}\left(e^{2\pi t}-1\right)}\,\mathrm{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{{Citation needed|date=October 2015}}

:<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 |accessdate=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]]<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 only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed 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 = http://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 /> but 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| bibcode = 2016arXiv160602044B}}</ref>
has found that this latter series was actually first published by [[Joseph Ser]] in 1926.<ref>{{cite journal|first = Joseph|last = Ser|authorlink = 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 = French}}</ref>  New proofs for both of these results were offered by Demetrios Kanoussis in 2017.<ref>{{cite journal|title = A New Proof of H. Hasse's Global Expression for the Riemann's Zeta Function|year = 2017|first = Demetrios P.|last = Kanoussis|url = https://www.researchgate.net/publication/317823796}}</ref> Other similar globally convergent series include

:<math>
\begin{align} 
\zeta(s) & =\frac{1}{s-1}\sum_{n=0}^\infty H_{n+1}\sum_{k=0}^n (-1)^k \binom{n}{k}(k+2)^{1-s} \\[6pt]
\zeta(s) & =\frac{1}{s-1}\left\{-1 + \sum_{n=0}^\infty H_{n+2}\sum_{k=0}^n (-1)^k \binom{n}{k}(k+2)^{-s}\right\} \\[6pt] 
\zeta(s) & =\frac{k!}{(s-k)_k}\sum_{n=0}^\infty \frac{1}{(n+k)!}\left[{n+k \atop n}\right]\sum_{\ell=0}^{n+k-1}\!(-1)^\ell \binom{n+k-1}{\ell} (\ell+1)^{k-s},\quad k=1, 2, 3,\ldots \\[6pt] 
\zeta(s) & =\frac{1}{s-1} + \sum_{n=0}^\infty |G_{n+1}| \sum_{k=0}^n(-1)^k \binom{n}{k}(k+1)^{-s} \\[6pt]
\zeta(s) & =\frac{1}{s-1}+1-\sum_{n=0}^\infty C_{n+1}\sum_{k=0}^n (-1)^k \binom{n}{k}(k+2)^{-s} \\[6pt] 
\zeta(s) & =\frac{2(s-2)}{s-1}\zeta(s-1) + 2\sum_{n=0}^\infty (-1)^n  G_{n+2}\sum_{k=0}^n (-1)^k \binom{n}{k} (k+1)^{-s} \\[6pt] 
\zeta(s) & =-\sum_{l=1}^{k-1} \frac{(k-l+1)_l}{(s-l)_l} \zeta(s-l) + \frac{k}{s-k}+k \sum_{n=0}^\infty (-1)^n G_{n+1}^{(k)}\sum_{k=0}^{n}(-1)^k \binom{n}{k} (k+1)^{-s} \\[6pt]
\zeta(s) & = \frac{(a+1)^{1-s} }{s-1} +  \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a)
\sum_{k=0}^n (-1)^k \binom{n}{k} (k+1)^{-s}  ,\quad \Re(a)>-1 \\[6pt]
\zeta(s) & =1 + \frac{(a+2)^{1-s}}{s-1} +  \sum_{n=0}^\infty (-1)^n \psi_{n+1}(a)
\sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+2)^{-s}  ,\quad \Re(a)>-1 \\[6pt] 
\zeta(s) & = \frac{1}{a+\tfrac{1}{2}}\left\{-\frac{\zeta(s-1,1+a)}{s-1} + \zeta(s-1) +  
 \sum_{n=0}^\infty (-1)^n \psi_{n+2}(a) \sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+1)^{-s}\right\} ,\quad \Re(a)>-1
\end{align}
</math>

where {{math|''H''<sub>''n''</sub>}} are the [[harmonic numbers]], <math>\left[{\cdot \atop \cdot}\right]</math> are the [[Stirling numbers of the first kind]], <math>(s-k)_k</math> is the [[Pochhammer symbol]], {{math|''G''<sub>''n''</sub>}} are the [[Gregory coefficients]],  {{math|''G''{{su|b=''n''|p=(''k'')}}}} are the [[Gregory coefficients]] of higher order, {{math|''C''<sub>''n''</sub>}} are the Cauchy numbers of the second kind ({{math|''C''<sub>1</sub> {{=}} 1/2}}, {{math|''C''<sub>2</sub> {{=}} 5/12}}, {{math|''C''<sub>3</sub> {{=}} 3/8}},...), and {{math|''&psi;<sub>n</sub>''(''a'')}}
are the [[Bernoulli polynomials of the second kind]], see Blagouchine's paper.<ref name="blag2018" />

[[Peter Borwein]] has 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|authorlink = 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}}</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
}}</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)}{s + 1} - \frac{1}{s (s + 1)}
  \end{align}</math>

== Numerical algorithms ==
For <math>v=1,2,3,\dots</math> , the Riemann zeta function has for fixed <math>\sigma_{0}<v</math> and for all <math>\sigma \leq \sigma_0</math> the following representation in terms of three absolutely and uniformly converging series,<ref name=":0">{{cite arxiv|last=Fischer|first=Kurt|date=2017-03-04|title=The Zetafast algorithm for computing zeta functions|eprint=1703.01414|class=math.NT}}</ref><math display="block">\begin{align}
\zeta\left(s\right) & =  \sum_{n=1}^{\infty}n^{-s}\sum_{w=0}^{v-1}\frac{\left(\frac{n}{N}\right)^{w}}{w!}e^{-\frac{n}{N}}-\frac{\Gamma\left(1-s+v\right)}{\left(1-s\right)\Gamma\left(v\right)}N^{1-s}+\sum_{\mu=\pm1}E_{\mu}\left(s\right)\\
E_{\mu}\left(s\right) & =  \left(2\pi\right)^{s-1}\Gamma\left(1-s\right)e^{i\mu\frac{\pi}{2}\left(1-s\right)}\sum_{m=1}^{\infty}\left[m^{s-1}-\sum_{w=0}^{v-1}\binom{s-1}{w}\left(m+\frac{i\mu}{2\pi N}\right)^{s-1-w}\left(\frac{-i\mu}{2\pi N}\right)^{w}\right]
\end{align}
</math>where for positive integer <math>s=k</math> one has to take the limit value  <math>\lim_{s\to k} E_\mu \left( s\right)</math>. The derivatives of <math>\zeta(s)</math> can be calculated by differentiating the above series termwise. From this follows an algorithm which allows to compute, to arbitrary precision, <math>\zeta(s)</math> and its derivatives using at most <math>C\left(\epsilon\right)\left|\tau\right|^{\frac{1}{2}+\epsilon}
</math> summands for any <math>\epsilon >0
</math>, with explicit error bounds. For <math>\zeta(s)</math>, these are as follows:

For a given argument <math>s
</math> with <math>0\leq\sigma\leq2
</math> and <math>0 < t
</math> one can approximate <math>\zeta(s)</math> to any accuracy <math>\delta\leq0.05
</math> by summing the first series to <math>n = \left\lceil 3.151 \cdot vN\right\rceil
</math>, <math>E_{1}\left(s\right)
</math> to <math>m=\left\lceil N\right\rceil 
</math> and neglecting <math>E_{-1}\left(s\right)
</math>, if one chooses <math>v
</math> as the next higher integer of the unique solution of <math>x-\max\left(\frac{1-\sigma}{2},0\right)\ln\left(\frac{1}{2}+x+\tau\right)=\ln\frac{8}{\delta}
</math> in the unknown <math>x
</math>, and from this <math>N=1.11\left(1+\frac{\frac{1}{2}+\tau}{v}\right)^{\frac{1}{2}}
</math>. For  <math>t=0
</math> one can neglect  <math>E_{1}\left(s\right)
</math> altogether. Under the mild condition <math>\tau>\frac{5}{3}\left(\frac{3}{2}+\ln\frac{8}{\delta}\right)
</math> one needs at most <math>2+8\sqrt{1+\ln\frac{8}{\delta}+\max\left(\frac{1-\sigma}{2},0\right)\ln\left(2\tau\right)}~\sqrt{\tau}
</math> summands. Hence this algorithm is essentially as fast as the [[Riemann-Siegel formula]]. Similar algorithms are possible for [[Dirichlet L-function]]s.<ref name=":0" />

==Applications==
The zeta function occurs in applied [[statistics]] (see [[Zipf's law]] and [[Zipf–Mandelbrot law]]).

[[Zeta function regularization]] is used as one possible means of [[regularization (physics)|regularization]] of [[divergent series]] and [[divergent integral]]s in [[quantum field theory]]. In one notable example, the Riemann
zeta-function shows up explicitly in one method of calculating the [[Casimir effect]]. The zeta function is also useful for the analysis of [[dynamical systems]].<ref>{{cite web|url=http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/spinchains.htm |title=Work on spin-chains by A. Knauf, et. al |website=Empslocal.ex.ac.uk |date= |accessdate=2017-01-04}}</ref>

===Infinite series===
The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.<ref>Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)</ref> 
*<math>\sum_{n=2}^\infty\bigl(\zeta(n)-1\bigr) = 1</math>

In fact the even and odd terms give the two sums
*<math>\sum_{n=1}^\infty\bigl(\zeta(2n)-1\bigr)=\frac{3}{4}</math>
and

*<math>\sum_{n=1}^\infty\bigl(\zeta(2n+1)-1\bigr)=\frac{1}{4}</math>
Parametrized versions of the above sums are given by

*<math>\sum_{n=1}^\infty(\zeta(2n)-1)\,t^{2n} = \frac{t^2}{t^2-1} + \frac{1}{2} \left(1- \pi t\cot(t\pi)\right)</math>
and

*<math>\sum_{n=1}^\infty(\zeta(2n+1)-1)\,t^{2n} = \frac{t^2}{t^2-1} + \frac{1}{2}\left(\psi^0(t)+\psi^0(-t) \right) - \gamma</math>

with <math>|t|<2</math> and where <math>\psi</math> and <math>\gamma</math> are the [[polygamma function]] and [[Euler's constant]], as well as

*<math>\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n}\,t^{2n} = \log\left(\dfrac{1-t^2}{\operatorname{sinc}(\pi\,t)}\right)</math>

all of which are continuous at <math>t=1</math>. Other sums include

*<math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n} = 1-\gamma</math>
*<math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n} \left(\left(\tfrac{3}{2}\right)^{n-1}-1\right) = \frac{1}{3} \ln \pi</math>
*<math>\sum_{n=1}^\infty\bigl(\zeta(4n)-1\bigr) = \frac78-\frac{\pi}{4}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right).</math>
*<math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n}\operatorname{Im}\bigl((1+i)^n-(1+i^n)\bigr) = \frac{\pi}{4}</math>

where {{math|Im}} denotes the [[imaginary part]] of a complex number.

There are yet more formulas in the article [[Harmonic number#Relation to the Riemann zeta function|Harmonic number.]]

==Generalizations==<!-- This section is linked from [[Power law]] -->
There are a number of related [[zeta function]]s that can be considered to be generalizations of the Riemann zeta function. These include the [[Hurwitz zeta function]]

:<math>\zeta(s,q) = \sum_{k=0}^\infty \frac{1}{(k+q)^s}</math>
(the convergent series representation was given by Helmut Hasse in 1930,<ref name = Hasse1930>{{Cite journal |first=Helmut |last=Hasse |authorlink=Helmut Hasse |title=Ein Summierungsverfahren für die Riemannsche {{mvar|ζ}}-Reihe |trans-title=A summation method for the Riemann ζ series |year=1930 |journal=[[Mathematische Zeitschrift]] |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 |language=German}}</ref>{{inconsistent citations}} cf. [[Hurwitz zeta function]]), which coincides with the Riemann zeta function when {{math|''q'' {{=}} 1}} (the lower limit of summation in the Hurwitz zeta function is 0, not 1), the [[Dirichlet L-function|Dirichlet {{mvar|L}}-functions]] and the [[Dedekind zeta-function]]. For other related functions see the articles [[zeta function]] and [[L-function|{{mvar|L}}-function]].

The [[polylogarithm]] is given by

:<math>\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}</math>

which coincides with the Riemann zeta function when {{math|''z'' {{=}} 1}}.

The [[Lerch transcendent]] is given by
:<math>\Phi(z, s, q) = \sum_{k=0}^\infty\frac { z^k} {(k+q)^s}</math>
which coincides with the Riemann zeta function when {{math|''z'' {{=}} 1}} and {{math|''q'' {{=}} 1}} (the lower limit of summation in the Lerch transcendent is&nbsp;0, not&nbsp;1).

The Clausen function {{math|Cl<sub>''s''</sub>(''θ'')}} that can be chosen as the real or imaginary part of {{math|Li<sub>''s''</sub>(''e''{{isup|''iθ''}})}}.

The [[multiple zeta functions]] are defined by

:<math>\zeta(s_1,s_2,\ldots,s_n) = \sum_{k_1>k_2>\cdots>k_n>0} {k_1}^{-s_1}{k_2}^{-s_2}\cdots {k_n}^{-s_n}.</math>

One can analytically continue these functions to the {{mvar|n}}-dimensional complex space. The special values taken by these functions at positive integer arguments are called [[multiple zeta values]] by number theorists and have been connected to many different branches in mathematics and physics.

==See also==
*[[1 + 2 + 3 + 4 + ···]]
*[[Arithmetic zeta function]]
*[[Generalized Riemann hypothesis]]
*[[Lehmer pair]]
*[[Particular values of Riemann zeta function]]
*[[Prime zeta function]]
*[[Riemann Xi function]]
*[[Renormalization]]
*[[Riemann–Siegel theta function]]
*[[ZetaGrid]]

==Notes==
{{Reflist|30em}}

==References==
*{{dlmf|id=25|first=T. M. |last=Apostol|title=Zeta and Related Functions}}
* {{cite journal|author1-link=Jonathan Borwein|first1=Jonathan|last1=Borwein |first2=David M. |last2=Bradley |author3-link=Richard Crandall|first3=Richard|last3=Crandall
| url = http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf| title = Computational Strategies for the Riemann Zeta Function| journal=J. Comp. App. Math.| year=2000| volume=121
| pages=247–296
|doi=10.1016/S0377-0427(00)00336-8
|issue=1–2|bibcode = 2000JCoAM.121..247B }}
* {{cite journal
|first1=Djurdje
|last1= Cvijović
|first2=Jacek
|last2= Klinowski
| title = Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments
| journal=J. Comp. App. Math.| year=2002| volume=142| pages=435–439
| doi = 10.1016/S0377-0427(02)00358-8
|mr=1906742
|issue=2|bibcode=2002JCoAM.142..435C}}
* {{cite journal
| first1=Djurdje
|last1= Cvijović
|first2= Jacek
|last2= Klinowski
| title = Continued-fraction expansions for the Riemann zeta function and polylogarithms| journal=Proc. Amer. Math. Soc.| year=1997| volume=125| pages=2543–2550| doi = 10.1090/S0002-9939-97-04102-6
| issue=9}}
* {{cite book|first=H. M. |last=Edwards|authorlink=Harold Edwards (mathematician)| title=Riemann's Zeta Function|url=https://archive.org/details/riemannszetafunc00edwa_0 |url-access=registration |publisher=Academic Press|year=1974|isbn=0-486-41740-9}} Has an English translation of Riemann's paper.
* {{cite journal | last1 = Hadamard | first1 = Jacques | authorlink = Jacques Hadamard | year = 1896 | title = Sur la distribution des zéros de la fonction {{math|''ζ''(''s'')}} et ses conséquences arithmétiques | url = | journal = Bulletin de la Société Mathématique de France | volume = 14 | issue = | pages = 199–220 | doi = 10.24033/bsmf.545 }}
* {{cite book|authorlink=G. H. Hardy|first=G. H.|last=Hardy|title=Divergent Series|publisher=Clarendon Press, Oxford|year=1949}}
* {{cite journal
|authorlink=Helmut Hasse
|first1=Helmut
|last1=Hasse
|title=Ein Summierungsverfahren für die Riemannsche {{mvar|ζ}}-Reihe
|year=1930
|journal=Math. Z.
|volume=32
|pages=458–464
|mr=1545177
|doi=10.1007/BF01194645
}} (Globally convergent series expression.)
* {{cite book|first=A. |last=Ivic|title=The Riemann Zeta Function|publisher=John Wiley & Sons|year=1985|isbn=0-471-80634-X}}
*{{cite book | first=Y. |last=Motohashi | title=Spectral Theory of the Riemann Zeta-Function | publisher= Cambridge University Press | year=1997| isbn=0521445205}}
* {{cite book|author1-link=A. A. Karatsuba|first1=A. A.|last1= Karatsuba |first2= S. M.|last2= Voronin|title=The Riemann Zeta-Function|publisher= W. de Gruyter|location= Berlin|year=1992}}
* {{cite journal
|first1=István
|last1=Mező
|first2=Ayhan
|last2=Dil
|doi=10.1016/j.jnt.2009.08.005
|title=Hyperharmonic series involving Hurwitz zeta function
|journal= Journal of Number Theory
|year=2010
|volume=130
|issue=2
|pages=360–369
|mr=2564902
}}
* {{cite book| first=Hugh L.|last= Montgomery| authorlink=Hugh Montgomery (mathematician)|first2=Robert C. |last2=Vaughan |authorlink2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory. I. Classical theory| series=Cambridge tracts in advanced mathematics| volume=97| publisher=Cambridge University Press| year=2007| isbn=978-0-521-84903-6|at=Ch. 10}}
* {{cite book| authorlink=Donald J. Newman| first1=Donald J. |last1=Newman| title=Analytic number theory| series=[[Graduate Texts in Mathematics]]| volume=177| publisher=Springer-Verlag| year=1998| isbn=0-387-98308-2 |at=Ch. 6}}
* {{cite journal
|first1=Guo
|last1=Raoh
|title=The Distribution of the Logarithmic Derivative of the Riemann Zeta Function
|journal=Proceedings of the London Mathematical Society
|year=1996
|volume=s3–72
|doi=10.1112/plms/s3-72.1.1 
|pages=1–27
|arxiv=1308.3597
}}
* {{Cite journal|first=Bernhard|last=Riemann|url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/|title=Über die Anzahl der Primzahlen unter einer gegebenen Grösse|year=1859|journal=Monatsberichte der Berliner Akademie}}. In ''Gesammelte Werke'', Teubner, Leipzig (1892), Reprinted by Dover, New York (1953).
* {{cite journal
|first1=Jonathan
|last1=Sondow
|doi=10.1090/S0002-9939-1994-1172954-7
|title= Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series
|journal=Proc. Amer. Math. Soc.
|year=1994
|pages=421–424
|issue=2
|volume=120
|url=http://www.ams.org/journals/proc/1994-120-02/S0002-9939-1994-1172954-7/S0002-9939-1994-1172954-7.pdf
}}
* {{cite book|authorlink=Edward Charles Titchmarsh|first=E. C.|last=Titchmarsh|title=The Theory of the Riemann Zeta Function |edition=2nd rev. |editor=Heath-Brown|publisher=Oxford University Press|year= 1986}}
* {{cite book|author1-link=E. T. Whittaker|first1=E. T.|last1=Whittaker|author2-link=G. N. Watson|first2=G. N.|last2=Watson|date=1927|title=A Course in Modern Analysis|edition=4th|publisher=Cambridge University Press|at=Ch. 13}}
* {{cite journal|first1=Jianqiang
|last1=Zhao
|doi=10.1090/S0002-9939-99-05398-8 
| title = Analytic continuation of multiple zeta functions| journal=Proc. Amer. Math. Soc.| year=1999| volume=128| pages=1275–1283|mr=1670846|issue=5
}}

==External links==
* {{springer|title=Zeta-function|id=p/z099260}}
* [http://mathworld.wolfram.com/RiemannZetaFunction.html Riemann Zeta Function, in Wolfram Mathworld] — an explanation with a more mathematical approach
* [http://dtc.umn.edu/~odlyzko/zeta_tables Tables of selected zeros]
* [http://seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php Prime Numbers Get Hitched] A general, non-technical description of the significance of the zeta function in relation to prime numbers.
* [https://arxiv.org/abs/math/0309433v1 X-Ray of the Zeta Function] Visually oriented investigation of where zeta is real or purely imaginary.
* [http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/ Formulas and identities for the Riemann Zeta function] functions.wolfram.com
* [http://www.math.sfu.ca/~cbm/aands/page_807.htm Riemann Zeta Function and Other Sums of Reciprocal Powers], section 23.2 of [[Abramowitz and Stegun]]
* {{cite web|last=Frenkel|first=Edward|author-link=Edward Frenkel|title=Million Dollar Math Problem|url=https://www.youtube.com/watch?v=d6c6uIyieoo|publisher=[[Brady Haran]]|accessdate=11 March 2014|format=video}}
* [http://www.mathematik.uni-stuttgart.de/~riedelmo/papers/rfeq.pdf Mellin transform and the functional equation of the Riemann Zeta function]—Computational examples of Mellin transform methods involving the Riemann Zeta Function

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