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| [[Image:Complex zeta.jpg|right|thumb|300px|Riemann zeta function ''ζ''(''s'') in the [[complex plane]]. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while [[hue]] encodes the value's [[Argument (complex analysis)|argument]].
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| The white spot at ''s'' = 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(''s'') = 1/2 are its zeros. Values with arguments close to zero including positive reals on the real half-line are presented in red.]]
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| The '''Riemann zeta function''' or '''Euler–Riemann zeta function''', ''ζ''(''s''), is a [[function (mathematics)|function]] of a [[complex variable]] ''s'' that [[analytic continuation|analytically continues]] the sum of the [[infinite series]] <math>\sum_{n=1}^\infty\frac{1}{n^s}</math>, which converges when the [[real part]] of ''s'' is greater than 1. More general [[Riemann_zeta#Representations|representations]] of ''ζ''(''s'') for all ''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]].
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| This function, as a function of a real argument, was introduced and studied by [[Leonhard Euler]] in the first half of the eighteenth century without using complex analysis, which was not available at that time. [[Bernhard Riemann]] in his memoir "[[On the Number of Primes Less Than a Given Magnitude]]" published in 1859 extended the Euler definition to a 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/millennium/Riemann_Hypothesis/riemann.pdf|title=The Riemann Hypothesis – official problem description|publisher=[[Clay Mathematics Institute]]|accessdate=2008-10-25}}</ref>
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| The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ''ζ''(2), provides a solution to the [[Basel problem]]. In 1979 [[Apéry]] proved the irrationality of [[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]]s and [[L-function]]s, are known.
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| ==Definition==
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| [[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.]]
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| The '''Riemann zeta function''' ''ζ''(''s'') is a function of a complex variable ''s'' = ''σ'' + ''it''. (The notation with ''s'', ''σ'', and ''t'' is traditionally used in the study of the ''ζ''-function, following Riemann.)
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| The following [[infinite series]] converges for all complex numbers ''s'' with real part greater than 1, and defines ''ζ''(''s'') in this case:
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| :<math>
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| \zeta(s) =
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| \sum_{n=1}^\infty n^{-s} =
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| \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\; \sigma = \mathfrak{R}(s) > 1.
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| \!</math>
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| The Riemann zeta function is defined as the [[analytic continuation]] of the function defined for σ > 1 by the sum of the preceding series.
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| [[Leonhard Euler]] considered the above series in 1740 for positive integer values of ''s'', and later [[Chebyshev]] extended the definition to real ''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>
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| The above series is a prototypical [[Dirichlet series]] that [[absolute convergence|converges absolutely]] to an [[analytic function]] for ''s'' such that {{nowrap|''σ'' > 1}} and [[divergent series|diverges]] for all other values of ''s''. Riemann showed that the function defined by the series on the half-plane of convergence can be [[analytic continuation|continued analytically]] to all complex values {{nowrap|''s'' ≠ 1}}. For ''s'' = 1 the series is the [[harmonic series (mathematics)|harmonic series]] which diverges to [[Extended real number line|+∞]], and
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| : <math> \lim_{s\to 1}(s-1)\zeta(s)=1.</math>
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| Thus the Riemann zeta function is a [[meromorphic function]] on the whole complex ''s''-plane, which is [[holomorphic function|holomorphic]] everywhere except for a [[simple pole]] at ''s'' = 1 with [[Residue (complex analysis)|residue]] 1.
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| ==Specific values==
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| [[File:Zeta.png|thumb|300px|Riemann zeta function for real ''s'' > 1]]
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| {{main|Particular values of Riemann zeta function}}
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| For any positive even integer ''2n'':
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| :<math> \zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}</math>
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| where ''B''<sub>2''n''</sub> is a [[Bernoulli number]].
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| For negative integers, one has
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| :<math>\zeta(-n)=-\frac{B_{n+1}}{n+1}</math>
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| for {{nowrap|''n'' ≥ 1}}, so in particular ''ζ'' vanishes at the negative even integers because ''B''<sub>''m''</sub> = 0 for all odd ''m'' other than 1.
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| For odd positive integers, no such simple expression is known.
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| :<math>\zeta(-1) = -\frac{1}{12}</math>
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| :: gives a way to assign a finite result to the divergent series [[1 + 2 + 3 + 4 + · · ·]], which can be useful 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 | pages = 22 | isbn = 978-0-521-63303-1}}</ref>
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| :<math>\zeta(0) = -\frac{1}{2};\!</math>
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| :<math>\zeta(1/2) \approx -1.4603545\!</math> {{OEIS|id=A059750}}
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| :: This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.<ref>A J Kainz and U M Titulaer, ''An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations'', pp. 1855-1874, J. Phys. A: Mathem. and General, V 25, No 7, 1992</ref>
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| :<math>\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty;\!</math>
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| ::if we approach from numbers larger than 1. Then this is the [[harmonic series (mathematics)|harmonic series]]. But its principal value
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| :<math> \lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}</math>
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| :exists which is the [[Euler–Mascheroni constant]] <math>\gamma = 0.5772\ldots</math>.
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| :<math>\zeta(3/2) \approx 2.612;\!</math> {{OEIS|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.
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| :<math>\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.645;\!</math> {{OEIS|id=A013661}}
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| :: 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>[[C. Stanley Ogilvy|C. S. Ogilvy]] & J. T. Anderson ''Excursions in Number Theory'', pp. 29–35, Dover Publications Inc., 1988 ISBN 0-486-25778-9</ref>
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| :<math>\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.202;\!</math> {{OEIS|id=A002117}}
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| :: This is called [[Apéry's constant]].
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| :<math>\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.0823;\!</math> {{OEIS|id=A0013662}}
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| :: This appears when integrating [[Planck's law]] to derive the [[Stefan–Boltzmann law]] in physics.
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| ==Euler product formula==
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| 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]]
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| :<math>\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}},</math>
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| where, by definition, the left hand side is ''ζ''(''s'') and the [[infinite product]] on the right hand side extends over all prime numbers ''p'' (such expressions are called [[Euler product]]s):
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| :<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>
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| Both sides of the Euler product formula converge for 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 ''s'' = 1, diverges, Euler's formula (which becomes <math>\textstyle\prod_pp/(p-1)</math>) implies that [[Euclid's theorem|there are infinitely many primes]].<ref>Charles Edward Sandifer, ''How Euler did it'', The Mathematical Association of America, 2007, p. 193. ISBN 978-0-88385-563-8</ref>
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| The Euler product formula can be used to calculate the [[asymptotic density|asymptotic probability]] that ''s'' randomly selected integers are set-wise [[coprime]]. Intuitively, the probability that any single number is divisible by a prime (or any integer), ''p'' is 1/''p''. Hence the probability that ''s'' numbers are all divisible by this prime is 1/''p''<sup>''s''</sup>, and the probability that at least one of them is ''not'' is {{nowrap|1 − 1/''p''<sup>''s''</sup>}}. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors ''n'' and ''m'' if and only if it is divisible by ''nm'', an event which occurs with probability 1/(''nm'')). Thus the asymptotic probability that ''s'' numbers are coprime is given by a product over all primes,
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| : <math>\prod_p^\infty \left(1-\frac{1}{p^s}\right) = \left( \prod_p^\infty \frac{1}{1-p^{-s}} \right)^{-1} = \frac{1}{\zeta(s)}. </math>
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| (More work is required to derive this result formally.)<ref>{{cite journal|author=J. E. Nymann|title=On the probability that ''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>
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| == The functional equation ==
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| The Riemann zeta function satisfies the [[functional equation]] (known as the '''Riemann functional equation''' or '''Riemann's functional equation''')
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| :<math>
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| \zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)
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| \!,</math>
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| where Γ(''s'') is the [[gamma function]], which is an equality of meromorphic functions valid on the whole [[complex plane]]. This equation relates values of the Riemann zeta function at the points ''s'' and {{nowrap|1 − ''s''}}. The functional equation (owing to the properties of the sine function) implies that ''ζ''(''s'') has a simple zero at each even negative integer ''s'' = −2''n'' — these are known as the '''[[Triviality (mathematics)|trivial]] zeros''' of ''ζ''(''s''). For ''s'' an even positive integer, the product sin(''πs''/2)Γ(1−''s'') is [[regular function|regular]] and the functional equation relates the values of the Riemann zeta function at odd negative integers and even positive integers.
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| 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)
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| : <math>
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| \eta(s)= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} = (1-{2^{1-s}})\zeta(s).
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| </math> | |
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| Incidentally, this relation is interesting also because it actually exhibits ''ζ''(''s'') as a [[Dirichlet series]] (of the ''η''-function) which is [[convergent]] (albeit [[absolute convergence|non-absolutely]]) in the larger half-plane ''σ'' > 0 (not just ''σ'' > 1), up to an elementary factor.
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| Riemann also found a symmetric version of the functional equation, given by first defining
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| :<math>\xi(s) = \frac{1}{2}\pi^{-s/2}s(s-1)\Gamma\left(\frac{s}{2}\right)\zeta(s).\!</math> | |
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| The functional equation is then given by
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| :<math>\xi(s) = \xi(1 - s).\!</math>
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| (Riemann defined a similar but different [[Riemann_Ξ_function|function which he called ''ξ''(''t'')]].)
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| ==Zeros, the critical line, and the Riemann hypothesis==
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| {{main|Riemann hypothesis}}
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| [[File:Zero-free region for the Riemann zeta-function.svg|right|thumb|300px|Apart from the trivial zeros, the Riemann zeta function doesn't have any zero on the right of σ=1 and on the left of 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 σ = 1/2 and, according to the Riemann Hypothesis, they all lie on the line σ = 1/2.]]
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| [[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 ''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.]]
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| 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 sin(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 {''s'' ∈ '''C''' : 0 < Re(''s'') < 1}, which is called the '''critical strip'''. The [[Riemann hypothesis]], considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero ''s'' has Re(''s'') = 1/2. In the theory of the Riemann zeta function, the set {''s'' ∈ '''C''' : Re(''s'') = 1/2} is called the '''critical line'''. For the Riemann zeta function on the critical line, see [[Z function|Z-function]].
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| === The Hardy–Littlewood conjectures ===
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| In 1914, [[G. H. Hardy|Godfrey Harold Hardy]] proved that <math>\zeta\bigl(\tfrac{1}{2}+it\bigr)</math> has infinitely many zeros.
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| Hardy and [[John Edensor Littlewood]] formulated two conjectures on the density and distance between the zeros of <math>\zeta\bigl(\tfrac{1}{2}+it\bigr)</math> on intervals of large positive real numbers. In the following, <math>N(T)</math> is the total number of real zeros and <math>N_0(T)</math> the total number of zeros of odd order of the function <math>\zeta\bigl(\tfrac{1}{2}+it\bigr)</math> lying in the interval <math>(0,T]</math>.
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| # For any <math>\varepsilon > 0</math>, there exists a <math>T_0(\varepsilon) > 0</math> such that when <math>T \geq T_0(\varepsilon)</math> and <math>H=T^{0.25+\varepsilon}</math>, the interval <math>(T,T+H]</math> contains a zero of odd order.
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| # For any <math>\varepsilon > 0</math>, there exists a <math>T_0(\varepsilon) > 0</math> and <math>c_\varepsilon > 0</math> 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)</math> and <math>H=T^{0.5+\varepsilon}</math>.
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| These two conjectures opened up new directions in the investigation of the Riemann zeta function.
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| === Other results ===
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| 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 Re(''s'') = 1 line.<ref name="Diamond1982">{{citation|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> is that {{nowrap|''ζ''(''σ'' + i''t'')}} ≠ 0 whenever | ''t'' | ≥ 3 and
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| :<math>\sigma\ge 1-\frac{1}{57.54(\log{|t|})^{2/3}(\log{\log{|t|}})^{1/3}}.\!</math>
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| 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 of the Riemann hypothesis|consequences]] in the theory of numbers.
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| It is known that there are infinitely many zeros on the critical line. [[John Edensor Littlewood|Littlewood]] showed that if the sequence (γ<sub>''n''</sub>) contains the imaginary parts of all zeros in the [[upper half-plane]] in ascending order, then
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| :<math>\lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_n\right)=0.\!</math>
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| The [[critical line theorem]] asserts that a positive percentage of the nontrivial zeros lies on the critical line.
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| In the critical strip, the zero with smallest non-negative imaginary part is {{nowrap|1/2 + ''i''14.13472514...}} ({{OEIS2C|A058303}}). Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis Re(''s'') = 1/2. Furthermore, the fact that <math>\zeta(s)=\overline{\zeta(\overline{s})}</math> for all complex {{nowrap|''s'' ≠ 1}} implies that the zeros of the Riemann zeta function are symmetric about the real axis.
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| ==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]] μ(''n''):
| |
| :<math>
| |
| \frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}
| |
| \!</math>
| |
| for every complex number ''s'' with real part > 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 ''s'' is greater than 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.
| |
| | |
| ===Estimates of the maximum of the modulus of the zeta function===
| |
| Let the functions <math>F(T;H)</math> and <math>G(s_{0};\Delta)</math> be defined by the equalities
| |
| | |
| : <math> F(T;H) = \max_{|t-T|\le H}\bigl|\zeta\bigl(\tfrac{1}{2}+it\bigr)\bigr|,\quad G(s_{0};\Delta) = \max_{|s-s_{0}|\le\Delta}|\zeta(s)|. </math> | |
| | |
| Here <math>T</math> is a sufficiently large positive number, <math>0<H\ll \ln\ln T</math>, <math>s_{0} = \sigma_{0}+iT</math>, <math>\tfrac{1}{2}\le\sigma_{0}\le 1</math>, <math>0<\Delta < \tfrac{1}{3}</math>. Estimating the values <math>F</math> and <math>G</math> from below shows, how large (in modulus) values <math>\zeta(s)</math> can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip <math>0\le Re \ s\le 1</math>.
| |
| | |
| The case <math>H\gg \ln\ln T</math> was studied by [[Kanakanahalli Ramachandra|Ramachandra]]; the case <math>\Delta
| |
| > c</math>, where <math>c</math> is a sufficiently large constant, is trivial.
| |
| | |
| [[Anatolii Alexeevitch Karatsuba|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| issue=70:5| 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.| issue=68:8| year=2004}}</ref> in particular, that if the values <math>H</math> and <math>\Delta</math> exceed certain sufficiently small constants, then the estimates
| |
| | |
| : <math> F(T;H) \ge T^{- c_1},\quad G(s_0; \Delta) \ge T^{-c_2}, </math>
| |
| | |
| hold, where <math>c_1, c_2</math> are certain absolute constants.
| |
| | |
| ===The argument of the Riemann zeta-function===
| |
| The function <math>S(t) = \frac{1}{\pi}\arg{\zeta\bigl(\tfrac{1}{2}+it\bigr)}</math> is called the argument of the Riemann zeta function.
| |
| Here <math>\arg{\zeta\bigl(\tfrac{1}{2}+it\bigr)}</math> is the increment of an arbitrary continuous branch of <math>\arg\zeta(s)</math> along the broken line joining the points <math>2, 2+it</math> and <math>\tfrac{1}{2}+it</math>
| |
| There are some theorems on properties of the function <math>S(t)</math>. 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=60:3| 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.| issue=60:5| year=1996}}</ref> are the mean value theorems for <math>S(t)</math> and its first integral <math>S_1(t) = \int_0^t S(u)du</math> on intervals of the real line, and also the theorem claiming that every interval <math>(T,T+H]</math> for <math>H \ge T^{27/82+\varepsilon}</math> contains at least
| |
| | |
| : <math> H(\ln T)^{1/3}e^{-c\sqrt{\ln\ln T}} </math>
| |
| | |
| points where the function <math>S(t)</math> changes sign. Earlier similar results were obtained by [[Atle Selberg]] for the case
| |
| <math>H\ge T^{1/2+\varepsilon}</math>. | |
| | |
| ==Representations==
| |
| | |
| ===Mellin transform===
| |
| The [[Mellin transform]] of a function ''ƒ''(''x'') is defined as
| |
| | |
| :<math> \int_0^\infty f(x)x^{s-1}\, dx, </math>
| |
| | |
| in the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of ''s'' is greater than one, we have
| |
| | |
| :<math>\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1}\,dx, </math>
| |
| | |
| where Γ denotes the [[Gamma function]]. By modifying the contour, Riemann showed that
| |
| | |
| :<math>2\sin(\pi s)\Gamma(s)\zeta(s) =i\oint_C \frac{(-x)^{s-1}}{e^x-1}\,dx </math>
| |
| | |
| for all ''s'', where the contour C starts and ends at +∞ and circles the origin once.
| |
| | |
| We can also find expressions which relate to prime numbers and the [[prime number theorem]]. If π(''x'') is the [[prime-counting function]], then
| |
| | |
| :<math>\log \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,dx,</math>
| |
| | |
| for values with {{nowrap|Re(''s'') > 1}}.
| |
| | |
| A similar Mellin transform involves the Riemann prime-counting function ''J''(''x''), which counts prime powers ''p''<sup>''n''</sup> with a weight of 1/''n'', so that
| |
| | |
| : <math>J(x) = \sum \frac{\pi(x^{1/n})}{n}.</math>
| |
| | |
| Now we have
| |
| | |
| :<math>\log \zeta(s) = s\int_0^\infty J(x)x^{-s-1}\,dx. </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 π(''x'') can be recovered from it by [[Möbius inversion formula|Möbius inversion]].
| |
| | |
| ===Theta functions===
| |
| The Riemann zeta function can be given formally by a divergent Mellin transform
| |
| | |
| :<math>2\pi^{-s/2}\Gamma(s/2)\zeta(s) = \int_0^\infty \theta(it)t^{s/2-1}\,dt,</math>
| |
| | |
| in terms of [[Theta function|Jacobi's theta function]]
| |
| | |
| :<math>\theta(\tau)= \sum_{n=-\infty}^\infty \exp(\pi i n^2\tau).</math>
| |
| | |
| However this integral does not converge for any value of ''s'' and so needs to be regularized: this gives the following expression for the zeta function:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| & {}\quad \pi^{-s/2}\Gamma(s/2)\zeta(s) \\[6pt]
| |
| & = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(it)-t^{-1/2}\right)t^{s/2-1}\,dt + \frac{1}{2}\int_1^\infty (\theta(it)-1)t^{s/2-1}\,dt.
| |
| \end{align}
| |
| </math>
| |
| | |
| ===Laurent series===
| |
| The Riemann zeta function is [[meromorphic]] with a single [[pole (complex analysis)|pole]] of order one at
| |
| ''s'' = 1. It can therefore be expanded as a [[Laurent series]] about ''s'' = 1;
| |
| the series development then is
| |
| | |
| :<math>\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n \; (s-1)^n.</math>
| |
| | |
| The constants 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{(\log k)^n}{k}\right) - \frac{(\log m)^{n+1}}{n+1}\right]}.</math>
| |
| | |
| The constant term γ<sub>0</sub> is the [[Euler–Mascheroni constant]].
| |
| | |
| === Integral ===
| |
| For all <math>s\in\mathbb{C}\setminus\{1\}</math> the integral relation
| |
| : <math>\zeta(s) = \frac{2^{s-1}}{s-1}-2^s\!\int_0^{\infty}\!\!\!\frac{\sin(s\arctan t)}{(1+t^2)^\frac{s}{2}(\mathrm{e}^{\pi\,t}+1)}\,\mathrm{d}t,</math>
| |
| holds true, which may be used for a numerical evaluation of the zeta-function.<ref>[http://mo.mathematik.uni-stuttgart.de/kurse/kurs5/seite19.html Mathematik-Online-Kurs: Numerik – Numerische Integration der Riemannschen Zeta-Funktion]</ref>
| |
| | |
| ===Rising factorial===
| |
| Another series development using the [[Pochhammer symbol|rising factorial]] valid for the entire complex plane is
| |
| | |
| :<math>\zeta(s) = \frac{s}{s-1} - \sum_{n=1}^\infty \left(\zeta(s+n)-1\right)\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 ''x''<sup>''s''−1</sup>; that context gives rise to a series expansion in terms of the [[falling factorial]].
| |
| | |
| ===Hadamard product===
| |
| {{Other uses|Matrix multiplication#Hadamard product{{!}}Matrix multiplication}}
| |
| On the basis of [[Weierstrass factorization theorem|Weierstrass's factorization theorem]], [[Hadamard]] gave the [[infinite product]] expansion
| |
| | |
| :<math>\zeta(s) = \frac{e^{(\log(2\pi)-1-\gamma/2)s}}{2(s-1)\Gamma(1+s/2)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^{s/\rho},\!</math>
| |
| | |
| where the product is over the non-trivial zeros ''ρ'' of ''ζ'' and the letter γ again denotes the [[Euler–Mascheroni constant]]. A simpler [[infinite product]] expansion is
| |
| | |
| :<math>\zeta(s) = \pi^{s/2} \frac{\prod_\rho \left(1 - \frac{s}{\rho} \right)}{2(s-1)\Gamma(1+s/2)}.\!</math>
| |
| | |
| This form clearly displays the simple pole at ''s'' = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at ''s'' = ''ρ'' (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeroes, i.e. the factors for a pair of zeroes of the form ''ρ'' and 1 − ''ρ'' should be combined.)
| |
| | |
| ===Logarithmic derivative on the critical strip===
| |
| | |
| : <math>
| |
| {\pi \frac{dN}{dx} (x) = \frac{1}{2i}\frac{d}{dx}\bigl(\log(\zeta(1/2 + ix)) - \log(\zeta(1/2 - ix))\bigr)- \frac{2}{1+4x^2} - \sum_{n=0}^\infty \frac{2n + 1/2}{(2n + 1/2)^2 +x^2}}
| |
| </math> | |
| | |
| where <math> \frac{dN(x)}{dx} = \sum_\rho \delta (x-\rho) </math> is the density of zeros of ''ζ'' on the critical strip 0 < Re(''s'') < 1 (''δ'' is the [[Dirac delta distribution]], and the sum is over the nontrivial zeros ''ρ'' of ''ζ'').
| |
| | |
| ===Globally convergent series===
| |
| A globally convergent series for the zeta function, valid for all complex numbers ''s'' except {{nowrap|''s'' {{=}} 1 + <big>{{sfrac|2π''i{{hsp}}n''|log(2)}}</big>}} for some integer ''n'', was conjectured by [[Konrad Knopp]] and proved by [[Helmut Hasse]] in 1930 (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 (-1)^k {n \choose 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 rediscovered more than 60 years later (see Sondow, 1994).
| |
| | |
| 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 {n\choose k}\frac{(-1)^k}{(k+1)^{s-1}}</math>
| |
| in the same publication.
| |
| | |
| [[Peter Borwein]] has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of [[Chebyshev polynomial]]s, is described in the article on the [[Dirichlet eta function]].
| |
| | |
| ==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 the calculation of the [[Casimir effect]]. The zeta function is also useful for the analysis of dynamical systems.<ref>{{cite web|title=Dynamical systems and number theory |url=http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/spinchains.htm}}</ref>
| |
| | |
| ===Infinite series===
| |
| | |
| The zeta function evaluated at positive integers appears in infinite series representations of a number of constants.<ref>Unless otherwise noted, the formulas in this section are from § 4 of J. M. Borwein et al. (2000)</ref> There are more formulas in the article [[Harmonic_number#Relation_to_the_Riemann_zeta_function|Harmonic number.]]
| |
| :<math>
| |
| 1=\sum_{n=2}^{\infty}(\zeta(n)-1).
| |
| </math> In fact the even and odd terms give the two sums <math>
| |
| \sum_{n=1}^{\infty}(\zeta(2n)-1)=\tfrac34
| |
| </math> and <math>
| |
| \sum_{n=1}^{\infty}(\zeta(2n+1)-1)=\tfrac14.
| |
| </math>
| |
| | |
| :<math>
| |
| \log 2=\sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{n}.
| |
| </math>
| |
| | |
| :<math>
| |
| 1-\gamma=\sum_{n=2}^{\infty}\frac{\zeta(n)-1}{n}
| |
| </math> where γ is [[Euler's constant]].
| |
| | |
| :<math>
| |
| \log \pi=\sum_{n=2}^{\infty}\frac{(2(\tfrac32)^n-3)(\zeta(n)-1)}{n}.
| |
| </math>
| |
| | |
| :<math>
| |
| \frac{\pi}{4}=\sum_{n=2}^{\infty}\frac{\zeta(n)-1}{n}\mathfrak{I}((1+i)^n-(1+i^n))
| |
| </math> where <math>\mathfrak{I}</math> represents the [[imaginary part]] of a [[complex number]].
| |
| | |
| Some zeta series evaluate to more complicated expressions
| |
| | |
| :<math>
| |
| \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{2^{2n}} = \frac16.
| |
| </math>
| |
| | |
| :<math>
| |
| \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{4^{2n}} = \frac{13}{30}-\frac{\pi}{8}.
| |
| </math>
| |
| | |
| :<math>
| |
| \sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{8^{2n}} = \frac{61}{126}-\frac{\pi}{16}(\sqrt2+1).
| |
| </math>
| |
| | |
| :<math>
| |
| \sum_{n=1}^{\infty}(\zeta(4n)-1) = \frac78-\frac{\pi}{4}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right).
| |
| </math>
| |
| | |
| ==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>{{Cite journal |first=Helmut |last=Hasse |title=Ein Summierungsverfahren für die Riemannsche ''ζ''-Reihe |year=1930 |journal=[[Mathematische Zeitschrift]] |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}</ref> cf. [[Hurwitz zeta function]]), which coincides with the Riemann zeta function when ''q'' = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the [[Dirichlet L-function]]s and the [[Dedekind zeta-function]]. For other related functions see the articles [[Zeta function]] and [[L-function]].
| |
| | |
| The [[polylogarithm]] is given by
| |
| | |
| :<math>\mathrm{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}\!</math>
| |
| | |
| which coincides with the Riemann zeta function when ''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 ''z'' = 1 and ''q'' = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).
| |
| | |
| The Clausen function Cl<sub>''s''</sub>(''θ'') that can be chosen as the real or imaginary part of Li<sub>''s''</sub>(''e''<sup> ''iθ''</sup>).
| |
| | |
| 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 ''n''-dimensional complex space. The special values of these functions are called [[multiple zeta values]] by number theorists and have been connected to many different branches in mathematics and physics.
| |
| | |
| ==See also==
| |
| *[[Generalized Riemann hypothesis]]
| |
| *[[Riemann–Siegel theta function]]
| |
| *[[Prime zeta function]]
| |
| *[[1 + 2 + 3 + 4 + ···]]
| |
| *[[renormalization]]
| |
| *[[Arithmetic zeta function]]
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| ==References==
| |
| | |
| *{{dlmf|id=25|first=T. M. |last=Apostol|title=Zeta and Related Functions}}
| |
| * {{cite journal| author=[[Jonathan Borwein]], David M. Bradley, [[Richard Crandall]]
| |
| | url = http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf|format=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 }} {{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 }} {{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|author=H. M. Edwards|authorlink=Harold Edwards (mathematician)| title=Riemann's Zeta Function|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 | author-separator =, | year = 1896 | title = Sur la distribution des zéros de la fonction ''ζ''(''s'') et ses conséquences arithmétiques | url = | journal = Bulletin de la Societé Mathématique de France | volume = 14 | issue = | pages = 199–220 }}
| |
| * {{cite book|author=[[G. H. 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 ''ζ''-Reihe
| |
| |year=1930
| |
| |journal=Math. Z.
| |
| |volume=32
| |
| |pages=458–464
| |
| |mr=1545177
| |
| |doi=10.1007/BF01194645
| |
| }} (Globally convergent series expression.)''
| |
| * {{cite book|author=A. Ivic|title=The Riemann Zeta Function|publisher=John Wiley & Sons|year=1985|isbn=0-471-80634-X}}
| |
| *{{cite book | author=Y. Motohashi | title=Spectral Theory of the Riemann Zeta-Function | publisher= Cambridge University Press | year=1997| isbn=0521445205}}
| |
| * {{cite book|author1-link=A. A. Karatsuba|author1=A. A. Karatsuba |author2= S.M. Voronin|title=The Riemann Zeta-Function|publisher= W. de Gruyter, 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| author=Hugh L. Montgomery| authorlink=Hugh Montgomery (mathematician)| coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]]| title=Multiplicative number theory I. Classical theory| series=Cambridge tracts in advanced mathematics| volume=97| publisher=Cambridge University Press| year=2007| isbn=0-521-84903-9}} Chapter 10.
| |
| * {{cite book| authorlink=Donald J. Newman| author=Donald J. Newman| title=Analytic number theory| series=[[Graduate Texts in Mathematics|GTM]]| volume=177| publisher=Springer-Verlag| year=1998| isbn=0-387-98308-2}} Chapter 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
| |
| }}
| |
| * {{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).
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| * {{cite journal
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| |first1=Jonathan
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| |last1=Sondow
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| |doi=10.1090/S0002-9939-1994-1172954-7
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| |title= Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series
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| |journal=Proc. Amer. Math. Soc.
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| |year=1994
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| |pages=421–424
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| |issue=120
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| |volume=120
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| }}
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| * {{cite book|author=[[Edward Charles Titchmarsh|E. C. Titchmarsh]]|title=The Theory of the Riemann Zeta Function, Second revised ([[Heath-Brown]]) edition|publisher=Oxford University Press|year= 1986}}
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| * [[E. T. Whittaker]] and [[G. N. Watson]] (1927). ''A Course in Modern Analysis'', fourth edition, Cambridge University Press (Chapter XIII).
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| * {{cite journal|first1=Jianqiang
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| |last1=Zhao
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| |doi=10.1090/S0002-9939-99-05398-8
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| | title = Analytic continuation of multiple zeta functions| journal=Proc. Amer. Math. Soc.| year=1999| volume=128| pages=1275–1283|mr=1670846|issue=5
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| }}
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| | |
| ==External links==
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| * {{springer|title=Zeta-function|id=p/z099260}}
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| * [http://mathworld.wolfram.com/RiemannZetaFunction.html Riemann Zeta Function, in Wolfram Mathworld] — an explanation with a more mathematical approach
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| * [http://dtc.umn.edu/~odlyzko/zeta_tables Tables of selected zeros]
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| * [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.
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| * [http://arxiv.org/abs/math/0309433v1 X-Ray of the Zeta Function] Visually oriented investigation of where zeta is real or purely imaginary.
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| * [http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/ Formulas and identities for the Riemann Zeta function] functions.wolfram.com
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| * [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]]
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| * [http://www.youtube.com/watch?v=MsBUTuYI62k The Riemann Hypothesis – A Visual Exploration] — a visual exploration of the Riemann Hypothesis and Zeta Function
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| {{L-functions-footer}}
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| {{DEFAULTSORT:Riemann Zeta Function}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Analytic number theory]]
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| [[Category:Meromorphic functions]]
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