View source for Riemann zeta function

{{Short description|Analytic function in mathematics}}

[[File:Cplot zeta.svg|right|thumb|250px|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|access-date=2017-01-04}}</ref>]]
[[File:Riemann-Zeta-Detail.png|right|thumb|200px|The pole at {{math|1=''z'' = 1}} and two zeros on the critical line]]
The '''Riemann zeta function''' or '''Euler–Riemann zeta function''', denoted by the lowercase [[Greek alphabet|Greek letter]] {{math|''ζ''}} ([[zeta]]), is a [[function (mathematics)|mathematical function]] of a [[complex variable]] defined as <math display="block"> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math> for <math>\mathrm{Re}(s) > 1</math>, and its [[analytic continuation]] elsewhere.<ref name=":0" />

The Riemann zeta function plays a pivotal role in [[analytic number theory]] and has applications in [[physics]], [[probability theory]], and applied [[statistics]].

[[Leonhard Euler]] first introduced and studied the function over the [[real numbers|reals]] in the first half of the eighteenth century. [[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]].  This paper also contained the [[Riemann hypothesis]], a [[conjecture]] about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in [[pure mathematics]].<ref>{{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]] | access-date=2014-08-08 | archive-date=22 December 2015 | archive-url=https://web.archive.org/web/20151222090027/http://www.claymath.org/sites/default/files/official_problem_description.pdf }}</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 [[Apéry's constant|{{math|''ζ''(3)}}]], and as a consequence, the number was named after him. 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. [[List of zeta functions|Many generalizations]] of the Riemann zeta function, such as [[Dirichlet series]], [[Dirichlet L-function|Dirichlet {{math|''L''}}-functions]] and [[L-function|{{math|''L''}}-functions]], are known.

== Definition ==
[[File:Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.pdf|thumb|upright|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|1=''s'' = ''σ'' + ''it''}}, where {{math|''σ''}} and {{math|''t''}} are real numbers. (The notation {{math|''s''}}, {{math|''σ''}}, and {{math|''t''}} is used traditionally in the study of the zeta function, following Riemann.) When {{math|1=Re(''s'') = ''σ'' > 1}}, the function can be written as a converging summation or as an 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) =\sum_{n=1}^\infty\frac{1}{n^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]]. The Riemann zeta function is defined for other complex values via [[analytic continuation]] of the function defined for {{math|''σ'' > 1}}.

[[Leonhard Euler]] considered the above series in 1740 for positive integer values of {{math|''s''}}, and later [[Chebyshev]] extended the definition to {{math|Re(''s'') > 1}}.<ref name='devlin'>{{cite book |last=Devlin |first=Keith |author-link=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 {{math|''s''}} such that {{math|''σ'' > 1}} and [[divergent series|diverges]] for all other values of {{math|''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|1=''s'' = 1}}, the series is the [[harmonic series (mathematics)|harmonic series]] which diverges to {{math|+∞}}, and
<math display="block"> \lim_{s \to 1} (s - 1)\zeta(s) = 1.</math>
Thus the Riemann zeta function is a [[meromorphic function]] on the whole complex plane, which is [[holomorphic function|holomorphic]] everywhere except for a [[simple pole]] at {{math|1=''s'' = 1}} with [[Residue (complex analysis)|residue]] {{math|1}}.

== Euler's product formula ==
In 1737, 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 {{math|''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|1=''s'' = 1}}, diverges, Euler's formula (which becomes {{math|Π<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> Since the logarithm of {{math|''p''/(''p'' − 1)}} is approximately {{math|1/''p''}}, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the [[sieve of Eratosthenes]] shows that the density of the set of primes within the set of positive integers is zero.

The Euler product formula can be used to calculate the [[asymptotic density|asymptotic probability]] that {{math|''s''}} randomly selected integers within a bound are set-wise [[coprime]]. Intuitively, the probability that any single number is divisible by a prime (or any integer) {{math|''p''}} is {{math|1/''p''}}. Hence the probability that {{math|''s''}} numbers are all divisible by this prime is {{math|1/''p''{{isup|''s''}}}}, and the probability that at least one of them is ''not'' is {{math|1 − 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 {{math|''n''}} and {{math|''m''}} [[if and only if]] it is divisible by&nbsp;{{math|''nm''}}, an event which occurs with probability&nbsp;{{math|1/(''nm'')}}). Thus the asymptotic probability that {{math|''s''}} numbers are coprime is given by a product over all primes,<ref>{{cite book
 | last = Mollin | first = Richard A.
 | isbn = 978-1-4200-8328-6
 | mr = 2560324
 | page = 220
 | publisher = CRC Press, Boca Raton, FL
 | series = Discrete Mathematics and its Applications (Boca Raton)
 | title = Advanced Number Theory with Applications
 | url = https://books.google.com/books?id=6I1setlljDYC&pg=PA220
 | year = 2010}}</ref>
: <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>

== Riemann's functional equation ==
This zeta function satisfies the [[functional equation]]
<math display="block"> \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 {{math|''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|1=''s'' = −2''n''}}, known as the '''[[Triviality (mathematics)|trivial]] zeros''' of {{math|''ζ''(''s'')}}. When {{math|''s''}} is an even positive integer, the product <math>\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)</math> 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. When {{math|''s''}} is {{math|0}}, the zero of the sine factor is cancelled by the simple pole of {{math|''ζ''(1)}}.

{{collapse top|title=Proof of Riemann's functional equation}}

A proof of the functional equation proceeds as follows:
We observe that if {{math|Re(''s'') > 0}}, then
<math display="block"> \int_0^\infty x^{ \frac{1}{2} s - 1 } e^{-n^2\pi x}\ \mathrm dx\ =\ \frac{\ \Gamma\!\left( \frac{s}{2} \right)\ }{\ n^s\ \pi^{\frac{s}{2}}\ } ~.</math>

As a result, if {{math|Re(''s'') > 1}} then
<math display="block"> \frac{\ \Gamma\!\left(\frac{s}{2}\right)\ \zeta(s)\ }{\ \pi^{ \frac{s}{2} }\ }\ =\ \sum_{n=1}^\infty\ \int_0^\infty\ x^{{s\over 2}-1}\ e^{-n^2 \pi x}\ \mathrm dx\ =\ \int_0^\infty x^{{s\over 2}-1} \sum_{n=1}^\infty e^{-n^2 \pi x}\ \mathrm dx\ ,</math>
with the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on <math>s</math>).

For convenience, let
<math display="block"> \psi(x)\ := \ \sum_{n=1}^\infty\ e^{-n^2 \pi x} ,</math>
which is a special case of the [[theta function]].

Because <math>e^{-n^2 \pi x}</math> and <math>\frac1\sqrt{x} e^{\frac{-n^2 \pi}{x}}</math> are [[Fourier transform#Definition|Fourier transform pairs]],<ref name='Damm-Johnsen'>{{cite book |last=Damm-Johnsenn |first=Håvard |title=Theta functions and their applications |year=2019 |page=5|url=https://users.ox.ac.uk/~quee4127/theta.pdf|archive-url=https://web.archive.org/web/20250321130227/https://users.ox.ac.uk/~quee4127/theta.pdf|archive-date=2025-03-21}}</ref> then, by the [[Poisson summation formula]], we have 
<math display="block"> \sum_{n=-\infty}^\infty\ e^{ - n^2 \pi\ x }\ =\ \frac{ 1 }{\ \sqrt{x\ }\ }\ \sum_{n=-\infty}^\infty\ e^{ -\frac{\ n^2 \pi\ }{ x } }\ ,</math>
so that
<math display="block">\ 2\ \psi(x) + 1\ =\ \frac{ 1 }{\ \sqrt{x\ }\ } \left(\ 2\ \psi\!\left( \frac{ 1 }{ x } \right) + 1\ \right) ~.</math>

Hence
<math display="block"> \pi^{ -\frac{s}{2} }\ \Gamma\!\left( \frac{s}{2} \right)\ \zeta(s)\ =\ \int_0^1\ x^{ \frac{s}{2} - 1 }\ \psi(x)\ \mathrm dx + \int_1^\infty x^{ \frac{s}{2} - 1 } \psi(x)\ \mathrm dx ~.</math>

The right side is equivalent to
<math display="block"> \int_0^1 x^{ \frac{s}{2} - 1 } \left( \frac{ 1 }{\ \sqrt{x\ }\ }\ \psi\!\left( \frac{1}{x} \right) + \frac{ 1 }{\ 2 \sqrt{x\ }\ } - \frac{ 1 }{ 2 }\ \right) \ \mathrm dx + \int_1^\infty x^{{s\over 2}-1} \psi(x)\ \mathrm dx </math>
or
<math display="block">
\frac{ 1 }{\ s - 1\ } - \frac{ 1 }{\ s\ } + \int_0^1\ x^{ \frac{s}{2} - \frac{3}{2}}\ \psi\!\left( \frac{ 1 }{\ x\ } \right)\ \mathrm dx + \int_1^\infty\ x^{ \frac{s}{2} - 1 }\ \psi(x)\ \mathrm dx
~.</math>

So
<math display="block">
\pi^{ -\frac{ s }{ 2 } }\ \Gamma\!\left( \frac{\ s\ }{ 2 } \right)\ \zeta(s)\ =\ \frac{ 1 }{\ s ( s - 1 )\ } + \int_1^\infty\ \left( x^{ -\frac{ s }{ 2 } - \frac{ 1 }{ 2 } } + x^{ \frac{ s }{ 2 } - 1 } \right)\ \psi(x)\ \mathrm dx
</math>
which is convergent for all {{math|''s''}}, because {{math|''ψ''(''x'') → 0}} more quickly than any power of {{math|''x''}} for {{math| ''x'' > 1}}, so the integral converges. As the RHS remains the same if {{math|''s''}} is replaced by {{math|1 − ''s''}},
<math display="block"> \frac{\ \Gamma\!\left(\ \frac{s}{2}\ \right)\ \zeta\!\left(\ s\ \right)\ }{\ \pi^{ \frac{s}{2}\ }\ }\ =\ \frac{\ \Gamma\!\left(\ \frac{1}{2} - \frac{s}{2}\ \right)\ \zeta\!\left(\ 1 - s\ \right)\ }{\ \pi^{ \frac{1}{2} - \frac{s}{2} }\ }  </math>
which is the functional equation attributed to [[Bernhard Riemann]].<ref>{{cite book |first=E.C. |last=Titchmarsh |year=1986 |title=The Theory of the Riemann Zeta Function |edition=2nd |publisher=Oxford Science Publications |place=[[Oxford]], UK |isbn=0-19-853369-1 |pages=21–22 }}</ref>

The functional equation above can be obtained using both the [[reflection formula]] and the [[Multiplication theorem#Gamma function–Legendre formula|duplication formula]].

First collect terms of {{math|''π''}}:
<math display="block">\Gamma\left(\frac{s}{2}\right)\zeta\left(s\right) = \Gamma\left(\frac{1}{2} - \frac{s}{2}\right)\zeta\left(1 - s\right)\pi^{s-\frac{1}{2}}</math>

Then multiply both sides by {{math|Γ(1 − ''s''/2)}} and use the reflection formula:
<math display="block">\Gamma\left(1-\frac s2\right)\Gamma\left(\frac{s}{2}\right)\zeta\left(s\right) = \Gamma\left(1-\frac s2\right)\Gamma\left(\frac{1}{2} - \frac{s}{2}\right)\zeta\left(1 - s\right)\pi^{s-\frac{1}{2}}</math>

<math display="block">\zeta\left(s\right) = \sin\left(\frac{\pi s}2\right)\Gamma\left(1-\frac s2\right)\Gamma\left(\frac{1}{2} - \frac{s}{2}\right)\zeta\left(1 - s\right)\pi^{s-\frac{3}{2}}</math>

Use the duplication formula with {{math|1=''z'' = (1 − ''s'')/2}}
<math display="block">\zeta\left(s\right) = \sin\left(\frac{\pi s}2\right)2^{1-1+s}\sqrt{\pi}\Gamma\left(1-s\right)\zeta\left(1 - s\right)\pi^{s-\frac{3}{2}}</math>
so that
<math display="block">\zeta\left(s\right) = \sin\left(\frac{\pi s}2\right)2^s\Gamma\left(1-s\right)\zeta\left(1 - s\right)\pi^{s-1}</math>

{{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.

In the adelic formulation developed in [[Tate's thesis]], the so-called Gamma factor
<math>\pi^{-s/2}\Gamma(s/2)</math> is interpreted as the local zeta factor at the [[Archimedean place]].

== Riemann's xi function ==
{{main|Riemann xi function}}

Riemann also found a [[Symmetry|symmetric]] version of the functional equation by setting
<math display="block">\xi(s) =\frac{s(s-1)}{2} \times \pi^{-\frac{s}{2}}\Gamma\left( \frac{s}{2} \right)\zeta(s) =  (s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}+1\right)\zeta(s)</math>
that satisfies:
<math display="block"> \xi(s) = \xi(1 - s) ~.</math>

Returning to the functional equation's derivation in the previous section, we have
<math display="block">
\xi(s) =\frac12 + \frac{s(s-1)}{2} \int_1^\infty \left(x^{-\frac{s}{2}-\frac{1}{2}} + x^{\frac{s}{2}-1}\right)\psi(x) dx
</math>

Using [[integration by parts]], 
<math display="block">
\xi(s) =\frac12 - \left[\left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi(x)\right]_1^\infty + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx
</math>
<math display="block">
\xi(s) =\frac12 + \psi(1) + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx
</math>

Using integration by parts again with a factorization of {{math|''x''<sup>3/2</sup>}},
<math display="block">
\xi(s) =\frac12 + \psi(1) - 2\left[x^{\frac32}\psi'(x)\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\right]_1^\infty + 2\int_1^\infty \left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\frac{d}{dx}\left[x^{\frac32}\psi'(x)\right] dx
</math>
<math display="block">
\xi(s) =\frac12 +\psi(1) + 4\psi'(1) + 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx
</math>

As <math>\frac12 +\psi(1) + 4\psi'(1)=0</math>,
<math display="block">
\xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx
</math>

Remove a factor of {{math|''x''<sup>−1/4</sup>}} to make the exponents in the remainder opposites. 
<math display="block">
\xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\left(x^{\frac{s-1/2}{2}} + x^{\frac{1/2-s}{2}}\right) dx
</math>

Using the [[hyperbolic functions]], namely {{math|1=cos(''x'') = cosh(''ix'')}}, and letting {{math|1=''s'' = 1/2 + ''it''}} gives
<math display="block">
\xi(s) = 4\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\cos(\frac{t}2\log x) dx
</math>
and by separating the integral and using the [[power series]] for {{math|cos}},
<math display="block">
\xi(s) = \sum_{n=0}^\infty a_{2n}t^{2n}
</math>
which led Riemann to his famous hypothesis.

== Zeros, the critical line, and the Riemann hypothesis ==
{{main|Riemann hypothesis}}
[[File:Zero-free region for the Riemann zeta-function.svg|right|thumb|300px|The Riemann zeta function has no zeros to the right of {{math|''σ'' {{=}} 1}} or (apart from the trivial zeros) to the left of {{math|1=''σ'' = 0}} (nor can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line {{math|1=''σ'' = 1/2}} and, according to the [[Riemann hypothesis]], they all lie on the line {{math|1=''σ'' = 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 {{math|''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 {{math|1=Re(''s'') = 1/2}}. The first non-trivial zeros can be seen at {{math|1=Im(''s'') =}} {{math|±14.135}}, {{math|±21.022}} and {{math|±25.011}}.]]
The functional equation shows that the Riemann zeta function has zeros at {{math|−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(''πs''/2)}} being {{math|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 important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {{math|{{mset|''s'' ∈ <math>\C</math> {{!}} 0 < Re(''s'') < 1}}}}, which is called the '''critical strip'''. The set {{math|{{mset|''s'' ∈ <math>\C</math> {{!}} Re(''s'') {{=}} 1/2}}}} is called the '''critical line'''. The [[Riemann hypothesis]], considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.<ref>{{cite journal | first = J. B. | last = Conrey | author-link = Brian Conrey | title = More than two fifths of the zeros of the Riemann zeta function are on the critical line | journal= J. Reine Angew. Math. | volume= 1989 | year = 1989 | issue = 399 | pages = 1–26 |url = http://www.digizeitschriften.de/resolveppn/GDZPPN002206781 | mr = 1004130 | doi = 10.1515/crll.1989.399.1 | s2cid = 115910600}}</ref> This has since been improved to 41.7%.<ref>{{cite journal | url=https://link.springer.com/article/10.1007/s40687-019-0199-8 | doi=10.1007/s40687-019-0199-8 | title=More than five-twelfths of the zeros of <math>\zeta </math> are on the critical line | date=2020 | last1=Pratt | first1=Kyle | last2=Robles | first2=Nicolas | last3=Zaharescu | first3=Alexandru | last4=Zeindler | first4=Dirk | journal=Research in the Mathematical Sciences | volume=7 | arxiv=1802.10521 }}</ref>

For the Riemann zeta function on the critical line, see [[Z function|{{math|''Z''}}-function]].

{| class="wikitable"
|+ First few nontrivial zeros<ref>{{cite web|url=https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html|title=Riemann Zeta Function Zeros|author=[[Eric Weisstein]]|access-date=2021-04-24}}</ref><ref>{{cite web|url=https://www.lmfdb.org/zeros/zeta/|author=The L-functions and Modular Forms Database|title=Zeros of &zeta;(''s'')}}</ref>
|-
! Zero
|-
| {{math|1/2 ± 14.134725... ''i''}}
|-
| {{math|1/2 ± 21.022040... ''i''}}
|-
| {{math|1/2 ± 25.010858... ''i''}}
|-
| {{math|1/2 ± 30.424876... ''i''}}
|-
| {{math|1/2 ± 32.935062... ''i''}}
|-
| {{math|1/2 ± 37.586178... ''i''}}
|-
| {{math|1/2 ± 40.918719... ''i''}}
|}

=== Number of zeros in the critical strip ===
Let {{math|''N''(''T'')}} be the number of zeros of {{math|''ζ''(''s'')}} in the critical strip {{math|0 < Re(''s'') < 1}}, whose imaginary parts are in the interval {{math|0 < Im(''s'') < ''T''}}.
[[Timothy Trudgian]] proved that, if {{math|''T'' > ''e''}}, then<ref>{{cite journal | first=Timothy S. | last=Trudgian | title = An improved upper bound for the argument of the Riemann zeta function on the critical line II | journal = J. Number Theory | date = 2014 | volume = 134 | pages = 280–292 | doi = 10.1016/j.jnt.2013.07.017 | arxiv = 1208.5846}}</ref>
: <math> \left|N(T) - \frac{T}{2\pi} \log{\frac{T}{2\pi e}}\right| \leq 0.112 \log T + 0.278 \log\log T + 3.385 + \frac{0.2}{T}</math>.

=== Hardy–Littlewood conjectures ===
In 1914, [[G. H. Hardy]] proved that {{math|''ζ''({{sfrac|1|2}} + ''it'')}} has infinitely many real zeros.<ref>{{cite journal|first1 = G.H. |last1 = Hardy |title = Sur les zeros de la fonction ζ(s) |journal = Comptes rendus de l'Académie des Sciences | volume = 158 |publisher = [[French Academy of Sciences]]|year = 1914 |pages = 1012–1014}}</ref><ref>{{cite journal |last1=Hardy |first1=G. H. |last2=Fekete |first2=M. |last3=Littlewood |first3=J. E. |date=1921-09-01 |title=The Zeros of Riemann's Zeta-Function on the Critical Line |journal=Journal of the London Mathematical Society |pages=15–19 |url=https://zenodo.org/record/1447415 |volume=s1-1 |doi=10.1112/jlms/s1-1.1.15 }}</ref>

Hardy and [[John Edensor Littlewood|J. E. Littlewood]] formulated two conjectures on the density and distance between the zeros of {{math|''ζ''(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''(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 number theory. The [[prime number theorem]] is equivalent to the fact that there are no zeros of the zeta function on the line {{math|1=Re(''s'') = 1}}.<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=553–589 |mr=670132 |doi=10.1090/S0273-0979-1982-15057-1 |doi-access=free }}</ref> It is also known that zeros do not exist in certain regions slightly to the left of the line {{math|1=Re(''s'') = 1}}, known as zero-free regions. For instance, Korobov<ref>{{cite journal | first1 =  Nikolai Mikhailovich| last1 = Korobov  | title = Estimates of trigonometric sums and their applications | journal = Usp. Mat. Nauk | volume = 13 | number = 4 | year = 1958 | pages=185–192  }}</ref> and Vinogradov<ref>{{cite journal | first1 = I.M.| last1 = Vinogradov  | title = Eine neue Abschätzung der Funktion {{nowrap|''ζ''(1 + ''it'')}} | journal = Russian. Izv. Akad. Nauk SSSR, Ser. Mat | volume = 22 | year = 1958 | pages =161–164  }}</ref> independently showed via the [[Vinogradov's mean-value theorem]] that for sufficiently large {{math|{{abs|''t''}}}}, {{math|''ζ''(''σ'' + ''it'') ≠ 0}} for
: <math>\sigma \geq 1 - \frac{c}{(\log|t|)^{2/3 + \varepsilon}}</math>
for any {{math|''ε'' > 0}} and a number {{math|''c'' > 0}} depending on {{math|''ε''}}. Asymptotically, this is the largest known zero-free region for the zeta function.

Explicit zero-free regions are also known. Platt and Trudgian<ref>{{cite journal | first1 =  David| last1 = Platt  | first2= Timothy S. | last2= Trudgian | title = The Riemann hypothesis is true up to 3⋅10<sup>12</sup> | journal = Bulletin of the London Mathematical Society| volume = 53 | number = 3 | year = 2021 | pages =792–797 | doi = 10.1112/blms.12460 | arxiv = 2004.09765}}</ref>
verified computationally that {{math|''ζ''(''σ'' + ''it'') ≠ 0}} if {{math|''σ'' ≠ 1/2}} and {{math|{{abs|''t''}} ≤ 3⋅10<sup>12</sup>}}. Mossinghoff, Trudgian and Yang proved<ref>{{cite journal | first1 = Michael J. | last1 = Mossinghoff | first2 = Timothy S. | last2 = Trudgian |first3 = Andrew | last3 = Yang | title = Explicit zero-free regions for the Riemann zeta-function | journal = Res. Number Theory | volume = 10 | year = 2024 | article-number = 11 | arxiv = 2212.06867 | doi = 10.1007/s40993-023-00498-y }}</ref> that zeta has no zeros in the region
: <math>\sigma\ge 1 - \frac{1}{5.558691\log|t|}</math>
for {{math|{{abs|''t''}} ≥ 2}}, which is the largest known zero-free region in the critical strip for {{math|3⋅10<sup>12</sup> < {{abs|''t''}} < exp(64.1) ≈ 7⋅10<sup>27</sup>}} (for previous results see<ref>{{cite journal | first1 = Michael J. | last1 = Mossinghoff | first2 = Timothy S. | last2 = Trudgian | title = Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function | journal = J. Number Theory | volume = 157 | year = 2015 | pages = 329–349 | arxiv = 1410.3926 | doi = 10.1016/J.JNT.2015.05.010| s2cid = 117968965 }}</ref>).
Yang<ref>{{cite journal | first1 =  Andrew| last1 = Yang | title =Explicit bounds on <math>\zeta(s)</math> in the critical strip and a zero-free region | journal = J. Math. Anal. Appl.| volume = 534 | number = 2 | year = 2024 | article-number =128124  | doi=10.1016/j.jmaa.2024.128124 | arxiv = 2301.03165 }}</ref> showed that {{math|''ζ''(''σ'' + ''it'') ≠ 0}} if
: <math>\sigma \geq 1 - \frac{\log\log|t|}{21.233\log|t|}</math> and <math>|t| \geq 3</math>
which is the largest known zero-free region for {{math|exp(170.2) < {{abs|''t''}} < exp(4.8⋅10<sup>5</sup>)}}.
Bellotti proved<ref>{{cite journal | first1 =  Chiara| last1 = Bellotti | title =Explicit bounds for the Riemann zeta function and a new zero-free region | journal = J. Math. Anal. Appl.| volume = 536 | number = 2 | year = 2024 | article-number =128249 | doi = 10.1016/j.jmaa.2024.128249 | arxiv = 2306.10680}}</ref> (building on the work of Ford<ref>{{cite journal | last1 = Ford | first1 = K. | year = 2002 | title = Vinogradov's integral and bounds for the Riemann zeta function | journal = Proc. London Math. Soc. | volume = 85 | issue = 3| pages = 565–633 | doi = 10.1112/S0024611502013655 | arxiv = 1910.08209 | s2cid = 121144007 }}</ref>) the zero-free region
: <math>\sigma \ge 1 - \frac{1}{53.989(\log|t|)^{2/3}(\log\log|t|)^{1/3}}</math> and <math>|t| \ge 3</math>.
This is the largest known zero-free region for fixed {{math|{{abs|''t''}} ≥ exp(4.8⋅10<sup>5</sup>)}}. Bellotti also showed that for sufficiently large {{math|{{abs|''t''}}}}, the following better result is known: {{math|''ζ''(''σ'' + ''it'') ≠ 0}} for
: <math>\sigma \geq 1 - \frac{1}{48.0718(\log|t|)^{2/3}(\log\log|t|)^{1/3}}.</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 {{math|1}}.)

In the critical strip, the zero with smallest non-negative imaginary part is {{math|1/2 + 14.13472514... ''i''}} ({{OEIS2C|A058303}}). The fact that, for all complex {{math|''s'' ≠ 1}},
: <math>\zeta(s)=\overline{\zeta(\overline{s})}</math>
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|1=Re(''s'') = 1/2}}.

It is also known that no zeros lie on the line with real part {{math|1}}.

A large class of modified zeta functions exists that share the same non-trivial zeros as the Riemann zeta function, where modification means replacing the prime numbers in the Euler product by real numbers, which was shown in a [[Grosswald–Schnitzer theorem|result by Grosswald and Schnitzer]].

== Specific values ==
{{main|Particular values of the Riemann zeta function}}
For any positive even integer {{math|2''n''}},
<math display="block"> \zeta(2n) = \frac{|{B_{2n}}|(2\pi)^{2n}}{2(2n)!},</math>
where {{math|''B''<sub>2''n''</sub>}} is the {{math|(2''n'')}}th [[Bernoulli number]].
The demonstration of the particular value
<math display="block">\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}</math>
is known as the [[Basel problem]]. The reciprocal of this sum answers the question: 'What is the probability that two numbers selected from a uniform distribution from {{math|1}} to {{math|''n''}}] are [[coprime]] as {{math|''n'' → ∞}}?'<ref>{{cite book|author-link=C. Stanley Ogilvy|first1=C. S. |last1=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>

For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic {{math|''K''}}-theory of the integers; see [[Special values of L-functions|Special values of {{math|''L''}}-functions]].
The value
<math display="block">\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.202056903159594285399...</math>
is [[Apéry's constant]].

For nonpositive integers, the series does not converge, but via [[analytic continuation]] one can show that
<math display="block">\zeta(-n)= -\frac{B_{n+1}}{n+1}</math>
for {{math|''n'' ≥ 0}} (using the convention that {{math|''B''<sub>1</sub> {{=}} 1/2}}).
In particular, {{math|''ζ''}} vanishes at the negative even integers because {{math|''B''<sub>''m''</sub> {{=}} 0}} for all odd {{math|''m''}} other than&nbsp;{{math|1}}. These are the so-called "trivial zeros" of the zeta function.
Another particular value is
<math display="block">\zeta(-1) = -\tfrac{1}{12}</math>
This gives a pretext for assigning a finite value to the divergent series {{math|[[1 + 2 + 3 + 4 + ⋯]]}}, which has been used in certain contexts ([[Ramanujan summation]]) such as [[string theory]].<ref name='polchinski'>{{cite book |last=Polchinski |first=Joseph |author-link=Joseph Polchinski |series=String Theory |volume=I |title=An Introduction to the Bosonic String |publisher=Cambridge University Press |year=1998 |page=22 |isbn=978-0-521-63303-1}}</ref>  Analogously,
<math display="block">\zeta(0) = -\tfrac{1}{2}</math>
can be viewed as assigning a finite result to the divergent series {{math|[[1 + 1 + 1 + 1 + ⋯]]}}.

The value
<math display="block">\zeta\bigl(\tfrac12\bigr) = -1.46035450880958681288\ldots</math>
is employed in calculating kinetic boundary layer problems of linear kinetic equations.<ref>{{cite journal|first1=A. J. |last1=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><ref>Further digits and references for this constant are available at {{OEIS2C|id=A059750}}.</ref>

Although
<math display="block">\zeta(1) = 1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots</math>
diverges, its [[Cauchy principal value]]
<math display="block"> \lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}</math>
exists and is equal to the [[Euler–Mascheroni constant]] {{math|1=''γ'' = 0.5772...}}.<ref name=Sondow1998>{{cite journal |last1=Sondow |first1=Jonathan |date=1998 |title=An antisymmetric formula for Euler's constant |journal=Mathematics Magazine |volume=71 |issue=3 |pages=219–220 |doi=10.1080/0025570X.1998.11996638 |access-date=2006-05-29 |url=http://home.earthlink.net/~jsondow/id8.html |archive-date=2011-06-04 |archive-url=https://web.archive.org/web/20110604123534/http://home.earthlink.net/~jsondow/id8.html}}</ref>

Taking the limit {{math|''s'' → +∞}} through the real numbers, one obtains {{math|1=''ζ''(+∞) = 1}}. But at [[complex infinity]] on the [[Riemann sphere]] the zeta function has an [[essential singularity]].<ref name=":0">{{cite journal |last1=Steuding |first1=Jörn |last2=Suriajaya |first2=Ade Irma |date=2020-11-01 |title=Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines |journal=Computational Methods and Function Theory |language=en |volume=20 |issue=3 |pages=389–401 |doi=10.1007/s40315-020-00316-x|s2cid=216323223 |issn=2195-3724|quote=Theorem 2 implies that ''ζ'' has an essential singularity at infinity |doi-access=free |hdl=2324/4483207 |hdl-access=free |arxiv=2007.14661 }}</ref>

== 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 {{math|''s''}} with real part greater than {{math|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 {{math|''s''}} is greater than {{math|1/2}}.

=== Universality ===
The critical strip of the Riemann zeta function has the remarkable property of '''universality'''. This [[zeta function universality]] states, roughly, that vertical translates of <math>\zeta(s)</math> can uniformly approximate any nonvanishing holomorphic function on suitable compact subsets of the strip <math>1/2<\operatorname{Re}(s)<1</math>.  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|Dirichlet {{math|''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 {{math|''T''}} is a sufficiently large positive number, {{math|0 < ''H'' ≪ log log ''T''}}, {{math|1=''s''<sub>0</sub> = ''σ''<sub>0</sub> + ''iT''}}, {{math|1/2 ≤ ''σ''<sub>0</sub> ≤ 1}}, {{math|0 < Δ < 1/3}}. Estimating the values {{math|''F''}} and {{math|''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 ''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 {{math|''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.

=== 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''(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|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 ''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>

== 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>
This recurrence leads to this other series development that uses the [[Pochhammer symbol|rising factorial]] and is valid for the entire complex plane <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 |title=A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator |url=http://linas.org/math/poch-zeta.pdf |access-date=2017-01-04 |website=Linas.org}}</ref>

=== 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. |author-link=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 {{math|''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 {{math|''s''}}<ref>Trivial exceptions of values of {{math|''s''}} that cause removable singularities are not taken into account throughout this article.</ref> (where {{math|''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|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 {{math|''s''}} is greater than {{math|1}}, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all {{math|''s''}} except {{math|0}} and {{math|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|1=''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 | url-access = subscription }}</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'' ∈ <math>\C</math>}}, {{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.

=== 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 {{math|''ρ''}} of {{math|''ζ''}} and the letter {{math|''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|1=''s'' = 1}}, the trivial zeros at {{math|−2, −4, }}... due to the gamma function term in the denominator, and the non-trivial zeros at {{math|1=''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 {{math|''ρ''}} and {{math|1 − ''ρ''}} should be combined.)

=== Globally convergent series ===
A globally convergent series for the zeta function, valid for all complex numbers {{math|''s''}} except {{math|1=''s'' = 1 + {{sfrac|2π''i''|ln 2}}''n''}} for some integer {{math|''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|1=''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''<sub>''n''</sub>}} are the Bernoulli numbers and {{math|(''x'')<sub>''k''</sub>}} 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|1=''s'' = 2, 4, 6, ...}}, 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''<sub>''k''</sub>}} 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}\left( k^{-3/4+\varepsilon }\right) \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}}</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 {{math|''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'' ∈ {{mset|−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'')|lh=1}}}} 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>

=== Mellin transform of the Engel map ===
The function {{math|1=''g''(''x'') = ''x''(1 + {{floor|''x''<sup>−1</sup>}}) − 1}} 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>

=== Stochastic representations ===
The [[Brownian motion and Riemann zeta function]] are connected through the [[moment-generating function]]s of [[stochastic process]]es derived from the [[Brownian motion]].<ref>{{cite journal
 | last1 = Biane | first1 = Philippe
 | last2 = Pitman | first2 = Jim
 | last3 = Yor | first3 = Marc
 | title = Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
 | journal = Bulletin of the American Mathematical Society |series=New Series
 | volume = 38 | issue = 4
 | pages = 435–465
 | year = 2001
 | doi = 10.1090/S0273-0979-01-00912-0 | doi-access = free
 | url = https://www.ams.org/journals/bull/2001-38-04/S0273-0979-01-00912-0/
 | publisher = American Mathematical Society
 | access-date = 2025-07-27
 }}</ref>

== Numerical algorithms ==
A classical algorithm, in use prior to about 1930, proceeds by applying the [[Euler–Maclaurin formula]] to obtain, for  positive integers {{math|''n''}} and {{math|''m''}},
: <math>\zeta(s) = \sum_{j=1}^{n-1}j^{-s} + \tfrac12 n^{-s} + \frac{n^{1-s}}{s-1} + \sum_{k=1}^m T_{k,n}(s) + E_{m,n}(s)</math>
where, letting <math>B_{2k}</math> denote the indicated [[Bernoulli number]],
: <math>T_{k,n}(s) = \frac{B_{2k}}{(2k)!} n^{1-s-2k}\prod_{j=0}^{2k-2}(s+j)</math>
and the error satisfies
: <math>|E_{m,n}(s)| < \left|\frac{s+2m+1}{\sigma + 2m + 1}T_{m+1,n}(s)\right|,</math>
with {{math|1=''σ'' = Re(''s'')}}.<ref>{{cite journal |mr=0961614
|last1=Odlyzko |first1= A. M. |author1-link=Odlyzko |last2= Schönhage |first2= A. |author2-link=Schönhage
|title=Fast algorithms for multiple evaluations of the Riemann zeta function
|journal=Trans. Amer. Math. Soc. |volume=309 |year=1988 |issue=2 |pages=797–809
|doi=10.2307/2000939 |doi-access=free |jstor=2000939 }}
</ref>

A modern numerical algorithm is the [[Odlyzko–Schönhage algorithm]].

== Applications ==
The zeta function occurs in applied [[statistics]] including [[Zipf's law]], [[Zipf–Mandelbrot law]], and [[Lotka's 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 |access-date=2017-01-04 }}</ref>

=== Musical tuning ===
In the theory of [[musical tuning]]s, the zeta function can be used to find [[Equal temperament|equal divisions of the octave]] (EDOs) that closely approximate the intervals of the [[Harmonic series (music)|harmonic series]]. For increasing values of <math>t \in \mathbb{R}</math>, the value of
: <math>\left\vert \zeta \left( \frac{1}{2} + \frac{2\pi{i}}{\ln{(2)}}t \right) \right\vert</math>
peaks near integers that correspond to such EDOs.<ref>{{cite web |url=https://oeis.org/A117536 |title=Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i×2×Pi/log(2)×t)) for increasing real t |author=Gene Ward Smith |website=The On-Line Encyclopedia of Integer Sequences |access-date=2022-03-04}}</ref> Examples include popular choices such as 12, 19, and 53.<ref>{{cite book |title=Tuning, Timbre, Spectrum, Scale |author=William A. Sethares |date=2005 |edition=2nd |publisher=Springer-Verlag London |page=74 |quote=... there are many different ways to evaluate the goodness, reasonableness, fitness, or quality of a scale ... Under some measures, 12-tet is the winner, under others 19-tet appears best, 53-tet often appears among the victors ...}}</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(\pi t)\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|{{abs|''t''}} < 2}} and where <math>\psi</math> and <math>\gamma</math> are the [[polygamma function]] and [[Euler's constant]], respectively, 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=1}^\infty\frac{\zeta(2n)-1}{n} = \ln 2</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}\Im \bigl((1+i)^n-1-i^n\bigr) = \frac{\pi}{4}</math>
where <math>\Im</math> denotes the [[imaginary part]] of a complex number.

Another interesting series that relates to the [[natural logarithm]] of the [[lemniscate constant]] is the following
* <math>\sum_{n=2}^\infty\left[\frac{2(-1)^n\zeta(n)}{4^n n}-\frac{(-1)^n\zeta(n)}{2^n n} \right]= \ln \left( \frac{\varpi}{2\sqrt2} \right)
</math>

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 |author-link=Helmut Hasse |title=Ein Summierungsverfahren für die Riemannsche ''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 |s2cid=120392534 |language=de}}</ref> 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 {{math|0}}, not {{math|1}}), the [[Dirichlet L-function|Dirichlet {{math|''L''}}-functions]] and the [[Dedekind zeta function]]. For other related functions see the articles [[zeta function]] and [[L-function|{{math|''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|1=''z'' = 1}}.
The [[Clausen function]] {{math|Cl<sub>''s''</sub>(''θ'')}} can be chosen as the real or imaginary part of {{math|Li<sub>''s''</sub>(''e''{{isup|''iθ''}})}}.

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|1=''z'' = 1}} and {{math|1=''q'' = 1}} (the lower limit of summation in the Lerch transcendent is&nbsp;{{math|0}}, not&nbsp;{{math|1}}).

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 {{math|''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]]
* [[Apéry's constant]]
* [[Basel problem]]
* [[Dirichlet eta function]]
* [[Generalized Riemann hypothesis]]
* [[Lehmer pair]]
* [[Particular values of the Riemann zeta function]]
* [[Prime zeta function]]
* [[Renormalization]]
* [[Riemann–Siegel theta function]]
* [[ZetaGrid]]

== References ==
{{reflist|25em}}

== Sources ==
{{refbegin|colwidth=25em|small=yes}}
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}}
* {{cite journal
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}}
* {{cite journal
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}}
* {{cite book
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}} Has an English translation of Riemann's paper.
* {{cite journal
 | last1 = Hadamard | first1 = Jacques | author-link = Jacques Hadamard
 | year = 1896
 | title = Sur la distribution des zéros de la fonction ''ζ''(''s'') et ses conséquences arithmétiques  |language=fr
 | trans-title = Regarding the distribution of the zeros of the function ''ζ''(''s'') and the arithmetical consequences
 | journal = Bulletin de la Société Mathématique de France
 | volume = 14 | pages = 199–220
 | doi = 10.24033/bsmf.545 | doi-access = free
}}
* {{cite book
 |first=G.H. |last=Hardy  |author-link=G. H. Hardy
 |year=1949
 |title=Divergent Series
 |publisher=Clarendon Press
 |place=Oxford, UK
}}
* {{cite journal
 |first = Helmut |last = Hasse  |author-link = Helmut Hasse
 |year=1930
 |title=Ein Summierungsverfahren für die Riemannsche ''ζ''-Reihe  |language=de
 |trans-title=A summation method for the Riemann ''ζ'' series
 |journal=Math. Z.
 |volume=32  |pages=458–464
 |mr=1545177 |s2cid=120392534
 |doi=10.1007/BF01194645
}} (Globally convergent series expression.)
* {{cite book
 |first=Aleksandar |last=Ivic | author-link=Aleksandar Ivić
 |year=1985
 |title=The Riemann Zeta Function
 |publisher=John Wiley & Sons
 |isbn=0-471-80634-X
}}
*{{cite book
 | first=Y. |last=Motohashi
 | year=1997
 | title=Spectral Theory of the Riemann Zeta-Function
 | publisher= Cambridge University Press
 | isbn=0-521-44520-5
}}
* {{cite book
 |first1=A.A. |last1= Karatsuba |author1-link=A. A. Karatsuba
 |first2=S.M. |last2= Voronin
 |year=1992
 |title=The Riemann Zeta-Function
 |publisher= W. de Gruyter
 |location= Berlin, DE
}}
* {{cite book
 | first1=Hugh L.   |last1= Montgomery |author-link=Hugh Montgomery (mathematician)
 | first2=Robert C. |last2=Vaughan     |author-link2=Robert Charles Vaughan (mathematician)
 | year=2007
 | title=Multiplicative Number Theory. I. Classical theory
 | series=Cambridge tracts in advanced mathematics  | volume=97
 | publisher=Cambridge University Press
 | isbn=978-0-521-84903-6
 | at=Chapter&nbsp;10
}}
* {{cite book
 | first1=Donald J. |last1=Newman   | author-link=Donald J. Newman
 | year=1998
 | title=Analytic Number Theory
 | series=[[Graduate Texts in Mathematics]]  | volume=177
 | publisher=Springer-Verlag
 | isbn=0-387-98308-2
 |at=Ch. 6
}}
* {{cite journal
 |first1=Guo |last1=Raoh
 |year=1996
 |title=The distribution of the logarithmic derivative of the Riemann zeta function
 |journal=Proceedings of the London Mathematical Society
 |volume=S3–72 |pages=1–27
 |doi=10.1112/plms/s3-72.1.1
}}
* {{cite journal
 |first=Bernhard  |last=Riemann |author-link=Bernhard Riemann
 |year=1859
 |title=Über die Anzahl der Primzahlen unter einer gegebenen Grösse
 |language=de, en
 |journal=Monatsberichte der Berliner Akademie
 |url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/
 |via=[[Trinity College, Dublin]] (maths.tcd.ie)
}} Also available in {{cite book
 |first=Bernhard  |last=Riemann |author-link=Bernhard Riemann
 |orig-date=1892  |year=1953
 |title=Gesammelte Werke |language=de
 |trans-title=Collected Works
 |editor=
 |edition=reprint
 |publisher=Dover (1953) / Teubner (1892)
 |place= New York, NY / Leipzig, DE
}}
* {{cite journal
 |first=Jonathan |last=Sondow
 |year=1994
 |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]]
 |volume=120  |issue=2  |pages=421–424
 |doi=10.1090/S0002-9939-1994-1172954-7  |doi-access=free
 |url=https://www.ams.org/journals/proc/1994-120-02/S0002-9939-1994-1172954-7/S0002-9939-1994-1172954-7.pdf
 |via=[[American Mathematical Society]] (ams.org)
}}
* {{cite book
 |first=E.C. |last=Titchmarsh  |author-link=Edward Charles Titchmarsh
 |year= 1986
 |title=The Theory of the Riemann Zeta Function |edition=2nd rev.
 |editor-last=Heath-Brown
 |publisher=Oxford University Press
}}
* {{cite book
 |first1=E.T. |last1=Whittaker  |author1-link=E. T. Whittaker
 |first2=G.N. |last2=Watson     |author2-link=G. N. Watson
 |date=1927
 |title=A Course in Modern Analysis |edition=4th
 |publisher=Cambridge University Press
 |at=Chapter&nbsp;13
}}
* {{cite journal
 | first =Jianqiang |last =Zhao
 | year=1999
 | title = Analytic continuation of multiple zeta functions
 | journal=[[Proceedings of the American Mathematical Society]]
 | volume=128 | issue=5 | pages=1275–1283
 | doi=10.1090/S0002-9939-99-05398-8  | doi-access=free
 | mr=1670846
}}
{{refend}}

== External links ==
*{{Commons category-inline}}
* {{springer|title=Zeta-function|id=p/z099260|mode=cs1}}
* [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] {{Webarchive|url=https://web.archive.org/web/20090517003700/http://dtc.umn.edu/~odlyzko/zeta_tables/ |date=17 May 2009 }}
* [https://web.archive.org/web/20080721030342/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|date=11 March 2014 |url=https://www.youtube.com/watch?v=d6c6uIyieoo |archive-url=https://ghostarchive.org/varchive/youtube/20211211/d6c6uIyieoo| archive-date=2021-12-11 |url-status=live|publisher=[[Brady Haran]]|access-date=11 March 2014|format=video}}{{cbignore}}
* [https://combinatorialsums.risc.jku.at/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
* [https://www.youtube.com/watch?v=sD0NjbwqlYw Visualizing the Riemann zeta function and analytic continuation] a video from [[3Blue1Brown]]

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