Riemann zeta function

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Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): strong colors denote values close to zero and hue encodes the value's argument. 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.

In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.

Riemann zeta function for real s > 1

The Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:

$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$

for certain values of s and then analytically continued to all complex s ≠ 1. This Dirichlet series converges for all real values of s greater than one. Since the 1859 paper of Bernhard Riemann, it has become standard to extend the definition of ζ(s) to complex values of the variable s, in two stages. First, Riemann showed that the series converges for all complex s whose real part Re(s) is greater than one and defines an analytic function of the complex variable s in the region {s ∈ C : Re(s) > 1} of the complex plane C. Secondly, he demonstrated how to extend the function ζ(s) to all complex values of s different from 1. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region {s∈C:s≠ 1} of the complex plane and has a simple pole at s=1. The analytic continuation process is unambiguous, resulting in a unique function, and in addition to extending ζ(s) beyond the domain of the convergence of the original series, Riemann established a functional equation for the zeta function, which relates its values at points s and 1 − s. The celebrated Riemann hypothesis, formulated in the same paper of Riemann, is concerned with zeros of this analytically extended function. To emphasize that s is viewed as a complex number, it is frequently written in the form s = σ + it, where σ = Re(s) is the real part of s and t = Im(s) is the imaginary part of s.

The connection between the zeta function and prime numbers was discovered by Leonhard Euler, who proved the identity

$\begin{align} \sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \\ & = \left(1 + \frac{1}{2^s} + \frac{1}{4^s} + \cdots \right) \left(1 + \frac{1}{3^s} + \frac{1}{9^s} + \cdots \right) \cdots \left(1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \cdots \right) \cdots, \end{align}$

where, by definition, the left hand side is ζ(s) and the infinite product in the right hand side extends over all prime numbers p (such expressions are called Euler products). Both sides of this identity converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula implies that there are infinitely many primes.

The above product can be used to show that $\frac 1{\zeta(n)}$ is the probability that n randomly selected integers are relatively prime.

For the Riemann zeta function on the critical line, see Z-function. For sums involving the zeta-function at integer values, see rational zeta series.

Main article: Zeta constant

The following are the most commonly used values of the Riemann zeta function.

$\zeta(0) = -1/2\,$

$\zeta(1/2) \approx -1.4603545088095868$

$\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty$ ; this is the harmonic series.

$\zeta(3/2) \approx 2.612$ ; this is employed in calculating the critical temperature for a Bose–Einstein condensate in Physics.

$\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.645$ ; the demonstration of this equality is known as the Basel problem.

$\zeta(5/2) \approx 1.341$

$\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.202$ ; this is called Apéry's constant

$\zeta(7/2) \approx 1.127$

$\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.0823$

The zeta-function satisfies the following functional equation:

$\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$

valid for all s in $\scriptstyle{C \setminus \lbrace 0,1 \rbrace}$ . Here, Γ denotes the gamma function. This formula, proved by Riemann in 1859, is used to construct the analytic continuation in the first place. (Actually, an equivalent relationship was conjectured by Euler in 1749 for the function $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}=\zeta(s)-\frac{2}{2^s}\zeta(s)$ . According to André Weil, Riemann seems to have been very familiar with Euler's work on the subject.^[1]) At s = 1, the zeta-function has a simple pole with residue 1. The equation also shows that the zeta function has trivial zeros at −2, −4, ... .

There is also a symmetric version of the functional equation, given by first defining

$\xi(s) = \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s).$

The functional equation is then given by

$\xi(s) = \xi(1 - s).\$

(Riemann defined a similar but different function which he called $ξ(t)$ .) The functional equation also gives the asymptotic limit

$\zeta \left( {1 - s} \right) = \left( {\frac{s}{{2\pi e}}} \right)^s \sqrt {\frac{{8\pi }}{s}} \cos \left( {\frac{{\pi s}}{2}} \right)\left( {1 + O\left( {\frac{1}{s}} \right)} \right).$

(Gergő Nemes, 2007)

The Riemann zeta function has zeros at the negative even integers (see the functional equation). These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/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, opens the way to an astonishingly rich vein of mathematical inquiry. 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 to be 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.

The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that all non-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result^[2] is that ζ(σ+it) ≠ 0 whenever |t| ≥ 3 and

$\sigma\ge 1-\frac{1}{57.54(\log{|t|})^{2/3}(\log{\log{|t|}})^{1/3}}.$

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

It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γ_n) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then

$\lim_{n\rightarrow\infty}\gamma_{n+1}-\gamma_n=0.$

The critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.

In the critical strip, the zero with smallest non-negative imaginary part is 1/2+i14.13472514... 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 ζ(s)=ζ(s*)* for all complex s ≠ 1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.

The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann Hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zeros has been studied using Rescaled Range Analysis^[3]. The self-similarity of the zero distributions is quite remarkable, and is characterized by a large fractal dimension of 1.9.

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infin} \frac{\mu(n)}{n^s}$

for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.

The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.

The Mellin transform of a function f(x) is defined as

$\{ \mathcal{M} f \}(s) = \int_0^\infty f(x)x^{s-1} \, dx$

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

$\Gamma(s)\zeta(s) =\left\{ \mathcal{M} \left(\frac{1}{\exp(x)-1}\right) \right\}(s),$

where Γ denotes the Gamma function. By subtracting off the first terms of the power series expansion of 1/(exp(x) − 1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we have

$\Gamma(s)\zeta(s) = \left\{ \mathcal{M}\left(\frac{1}{\exp(x)-1}-\frac1x\right)\right\}(s)$

and when the real part of s is between −1 and 0,

$\Gamma(s)\zeta(s) = \left\{\mathcal{M}\left(\frac{1}{\exp(x)-1}-\frac1x+\frac12\right)\right\}(s).$

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, then

$\log \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,dx$

for values with $\Re(s)>1$ . We can relate this to the Mellin transform of π(x) by $\frac{\log \zeta(s)}{s} - \omega(s) = \left\{\mathcal{M} \pi(x)\right\}(-s)$ where

$\omega(s) = \int_0^\infty \frac{\pi(s)}{x^{s+1}(x^s-1)}\,dx$

converges for $\Re(s)>\frac12$ .

A similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers pⁿ with a weight of 1/n, so that $J(x) = \sum \frac{\pi(x^{1/n})}{n}.$ Now we have

$\frac{\log \zeta(s)}{s} = \left\{\mathcal{M} J \right\}(-s).$

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.

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development then is

$\zeta(s) = \frac{1}{s-1} + \gamma_0 + \gamma_1(s-1) + \gamma_2(s-1)^2 + \cdots.$

The constants here are called the Stieltjes constants and can be defined as

$\gamma_k = \frac{(-1)^k}{k!} \lim_{N \rightarrow \infty} \left(\sum_{m \le N} \frac{\ln^k m}{m} - \frac{\ln^{k+1}N}{k+1}\right).$

The constant term γ₀ is the Euler-Mascheroni constant.

Another series development valid for the entire complex plane is

$\zeta(s) = \frac{1}{s-1} - \sum_{n=1}^\infty (\zeta(s+n)-1)\frac{s^{\overline{n}}}{(n+1)!}$

where $s^{\overline{n}}$ is the rising factorial $s^{\overline{n}} = s(s+1)\cdots(s+n-1)$ . 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^s−1; that context gives rise to a series expansion in terms of the falling factorial.

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

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

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler-Mascheroni constant.

A globally convergent series for the zeta function, valid for all complex numbers s except s = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930:

$\zeta(s)=\frac{1}{1-2^{1-s}} \sum_{n=0}^\infty \frac {1}{2^{n+1}} \sum_{k=0}^n (-1)^k {n \choose k} (k+1)^{-s}.$

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).

Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.

Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.

During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum 1 + 2 + 3 + 4 + · · ·, but we can re-write it as a sum of reciprocals:

$\begin{align} S &{}=1 + 2 + 3 + 4 + \cdots \\ &{}= \left(\frac{1}{1}\right)^{-1} + \left(\frac{1}{2}\right)^{-1} + \left(\frac{1}{3}\right)^{-1} + \left(\frac{1}{4}\right)^{-1} + \cdots \\ &{}=\sum_{n=1}^{\infin} \frac{1}{n^{-1}}. \end{align}$

The sum S appears to take the form of $ζ( - 1)$ . However, −1 lies outside of the domain for which the Dirichlet series for the zeta-function converges. However, a divergent series of positive terms such as this one can sometimes be summed in a reasonable way by the method of Ramanujan summation (see Hardy, Divergent Series.) Ramanujan summation involves an application of the Euler-Maclaurin summation formula, and when applied to the zeta-function, it extends its definition to the whole complex plane. In particular

$1+2+3+\cdots = -\frac{1}{12} (\Re)$

where the notation $(\Re)$ indicates Ramanujan summation^[4].

For even powers we have:

$1+2^{2k}+3^{2k}+\cdots = 0 (\Re)$

and for odd powers we have a relation with the Bernoulli numbers:

$1+2^{2k-1}+3^{2k-1}+\cdots = -\frac{B_{2k}}{2k} (\Re).$

Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.

There are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. These include the Hurwitz zeta function

$\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s},$

which coincides with Riemann's zeta-function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function and L-function.

The polylogarithm is given by

$\mathrm{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}$

which coincides with Riemann's zeta-function when z = 1.

The Lerch transcendent is given by

$\Phi(z, s, q) = \sum_{k=0}^\infty \frac { z^k} {(k+q)^s}$

which coincides with Riemann's 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 $C l s (θ)$ that can be chosen as the real or imaginary part of $Li s (e i θ)$

The multiple zeta functions are defined by

$\zeta(s_1,s_2,\dots,s_n) = \sum_{k_1>k_2>\dots>k_n>0} k_1^{-s_1}k_2^{-s_2}\cdots k_n^{-s_n}.$

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.

Neal Stephenson's 1999 novel Cryptonomicon mentions the zeta-function as a pseudo-random number source, a useful component in cipher design.

The zeta-function is a major part of the plot of Thomas Pynchon's novel Against the Day (2006).

The popular T.V. Show NUMB3RS had an episode ("Prime Suspect") in which criminals kidnapped a child and demanded as ransom a possible proof of the Riemann Hypothesis from a mathematician. The proof would be used to steal interest rates from an encrypted website.

^ "Euler and the Zeta Function" by Raymond Ayoub, American Mathematical Monthly, v. 81, pp. 1067-86, Dec. 1974
^ Ford, K. Vinogradov's integral and bounds for the Riemann zeta function, Proc. London Math. Soc. (3) 85 (2002), pp. 565-633
^ O. Shanker (2006). "Random matrices, generalized zeta functions and self-similarity of zero distributions". J. Phys. A: Math. Gen. 39: 13983-13997.
^ http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf

Riemann, Bernhard (1859), "Über die Anzahl der Primzahlen unter einer gegebenen Grösse", Monatsberichte der Berliner Akademie, <http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/>. In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953).
Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bulletin de la Societé Mathématique de France 14 (1896) pp 199-220.
Helmut Hasse, Ein Summierungsverfahren für die Riemannsche ζ-Reihe, (1930) Math. Z. 32 pp 458-464. (Globally convergent series expression.)
E. T. Whittaker and G. N. Watson (1927). A Course in Modern Analysis, fourth edition, Cambridge University Press (Chapter XIII).
H. M. Edwards (1974). Riemann's Zeta Function. Academic Press. ISBN 0-486-41740-9.
G. H. Hardy (1949). Divergent Series. Clarendon Press, Oxford.
A. Ivic (1985). The Riemann Zeta Function. John Wiley & Sons. ISBN 0-471-80634-X.
E. C. Titchmarsh (1986). The Theory of the Riemann Zeta Function, Second revised (Heath-Brown) edition. Oxford University Press.
Jonathan Borwein, David M. Bradley, Richard Crandall (2000). "Computational Strategies for the Riemann Zeta Function". J. Comp. App. Math. 121: p.11. (links to PDF file)

Djurdje Cvijović and Jacek Klinowski (2002). "Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments". J. Comp. App. Math. 142: pp.435-439.

Djurdje Cvijović and Jacek Klinowski (1997). "Continued-fraction expansions for the Riemann zeta function and polylogarithms". Proc. Amer. Math. Soc. 125: pp.2543-2550.

Jonathan Sondow, "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proc. Amer. Math. Soc. 120 (1994) 421-424.

Jianqiang Zhao (1999). "Analytic continuation of multiple zeta functions". Proc. Amer. Math. Soc. 128: pp.1275--1283.

Riemann Zeta Function, in Wolfram Mathworld - an explanation with more mathematical approach
Tables of selected zeroes
File with 1,000,000 zeros and accurate to about 60+ digits (To download compressed archive, click on Download Now... button.)
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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