NationMaster - Encyclopedia: Riemann zeta function

In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Bernhard Riemann. ... Partial plot of a function f. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time and explaining them using mathematics. ... It has been suggested that this article or section be merged with Probability axioms. ... Template:Otherusescccc A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ...

Definition

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: Image File history File links Download high resolution version (1024x715, 17 KB)Riemann Zeta function on [0. ... Image File history File links Download high resolution version (1024x715, 17 KB)Riemann Zeta function on [0. ... In mathematics, a series is a sum of a sequence of terms. ...

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

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 bigger 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, 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 and t=Im(s) is the imaginary part of s. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, the real numbers may be described informally in several different ways. ... On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. ... Bernhard Riemann. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f(z) = 1/zn at z = 0; a pole of a function f is a point a such that f(z) approaches infinity as z approaches a. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always Â½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

Relationship to prime numbers

The connection between this function and prime numbers was already realized by Leonhard Euler, who discovered In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 7, 1783) was a Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

$begin{align} zeta(s)& = prod_{pinmathbb{P}} frac{1}{1-p^{-s}} & = left(1 + frac{1}{2^s} + frac{1}{4^s} + frac{1}{8^s} + cdots right) left(1 + frac{1}{3^s} + frac{1}{9^s} + frac{1}{27^s} + cdots right) cdots left(1 + frac{1}{p^s} + frac{1}{p^{2s}} + frac{1}{p^{3s}} + cdots right) cdots, end{align}$

an infinite product extending over all prime numbers p. This Euler product formula converges for Re(s) > 1. It is a consequence of two simple and fundamental results in mathematics; the formula for the geometric series and the fundamental theorem of arithmetic. Euler's formula for ζ(s) is proved here. In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ... In mathematics, an Euler product is an infinite product expansion, indexed by prime numbers p, of a Dirichlet series. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... We will prove the that the following formula holds: where Î¶ denotes the Riemann zeta function and the product extends over all prime numbers p. ...

Various properties

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. In mathematics, the Z-function is a function used for studying the Riemann zeta-function along the critical line where the real part of the argument is one-half. ... In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. ...

Specific values

Main article: Zeta constant

The following are the most commonly used values of the Riemann zeta function. In mathematics, a zeta constant is a number obtained by plugging an integer into the Riemann zeta function. ...

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

$zeta(3/2) approx 2.612$

$zeta(2) = 1 + frac{1}{2^2} + frac{1}{3^2} + cdots = frac{pi^2}{6} approx 1.645$ ; 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

$zeta(7/2) approx 1.127$

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

See harmonic series (music) for the (related) musical concept. ... The Basel problem is a famous problem in number theory, first posed by Pietro Mengoli in 1644, and solved by Leonhard Euler in 1735. ... In mathematics, ApÃ©rys constant is a curious number that occurs in a variety of situations. ...

The functional equation

The zeta-function satisfies the following functional equation: In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...

$zeta(s) = 2^spi^{s-1}sinleft(frac{pi s}{2}right)Gamma(1-s)zeta(1-s)$

valid for all s in $scriptstyle{C setminus lbrace 0,1 rbrace}$ . Here, Γ denotes the gamma function. This formula is used to construct the analytic continuation in the first place. 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, ... . The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. ...

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

$&# 0;s) = pi^{-s/2}Gammaleft(frac{s}{2}right)zeta(s).$

The functional equation is then given by

$&# 0;s) = &# 0;1 - s).$

The functional equation also gives the asymptotic limit

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

(Nemes)

Zeros of the Riemann zeta function

The Riemann zeta function has zeros at the negative even integers (see the functional equation). These are called the trivial zeros. They are trivial only in the sense that their existence is relatively easy to prove, for example, from the connection with the gamma function as shown below. 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. Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always Â½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...

The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that at all non-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result^[1] is that ζ(σ+it) ≠ 0 whenever |t| ≥ 3 and In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

$sigmage 1-frac{1}{57.45(log{|t|})^{3/2}(log{log{|t|}})^{1/3}}.$

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. Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always Â½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...

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 John Edensor Littlewood (June 9, 1885 â€“ September 6, 1977) was a British mathematician. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...

$lim_{nrightarrowinfty}gamma_{n+1}-gamma_n=0.$

The critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line. In mathematics, the critical line theorem tells us that a positive percentage of the nontrivial zeros of the Riemann zeta function lie 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. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

Reciprocal

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n): In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... The classical MÃ¶bius function is an important multiplicative function in number theory and combinatorics. ...

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

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. In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...

The above, together with the expression for ζ(2), can be used to prove that the probability of two random integers being coprime is 6/π². The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2. Coprime - Wikipedia /**/ @import /skins-1. ...

Universality

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. In mathematics, the universality of zeta-functions refers to the remarkable property of the Riemann zeta-function, and other, similar functions, such as the Dirichlet L-functions, to approximate an arbitrary holomorphic function arbitrarily well. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Representations

Mellin transform

The Mellin transform of a function f(x) is defined as In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ...

${ mathcal{M} f }(s) = int_0^infty f(x)x^s frac{dx}{x}$

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

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

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

$Gamma(s)zeta(s) = left{ mathcal{M}left(frac{1}{exp(x)-1}-frac1xright)right}(s)$

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

$Gamma(s)zeta(s) = left{mathcal{M}left(frac{1}{exp(x)-1}-frac1x+frac12right)right}(s)$

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime counting function, then In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... In mathematics, the prime counting function is the function counting the number of primes less than or equal to some real number x. ...

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

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 classic MÃ¶bius inversion formula was introduced into number theory during the 19th century by August Ferdinand MÃ¶bius. ...

Laurent series

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 A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ... A Laurent series is defined with respect to a particular point c and a path of integration Î³. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...

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

The constants here are called the Stieltjes constants and can be defined as In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function: The Stieltjes constants can also be defined as the value of the limit The zeroth constant is known as the Euler-Mascheroni constant. ...

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

The constant term γ₀ is the Euler-Mascheroni constant. The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is Î³ â‰ˆ 0. ...

Rising factorial

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. In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...

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. In mathematics, the Gauss-Kuzmin-Wirsing operator occurs in the study of continued fractions; it is also related to the Riemann zeta function. ... In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ...

Hadamard product

On the basis of Weierstrass' factorization theorem, Hadamard gave the infinite product expansion In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes. ... Jacques Solomon Hadamard (December 8, 1865 - October 17, 1963) was a mathematician best known for his proof of the prime number theorem. ... In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ...

$zeta(s) = frac{e^{As}}{2(s-1)Gamma(1+s/2)} prod_rho left(1 - frac{s}{rho} right) e^{s/rho}$

where the product is over the non-trivial zeros ρ of ζ and

A = log(2π) − 1 − γ/2,

the letter γ again denoting the Euler-Mascheroni constant. The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is Î³ â‰ˆ 0. ...

Globally convergent series

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: Konrad Hermann Theodor Knopp (22 July 1882, Berlin, Germany â€“ 20 April 1957, Annecy, France) was a mathematician. ... Helmut Hasse (pronounced HAHS uh) (25 August 1898- 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry (Hasse principle), and to local zeta functions. ... Year 1930 (MCMXXX) was a common year starting on Wednesday (link is to a full 1930 calendar). ...

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

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. Peter B. Borwein is a Canadian mathematician, co-developer of an algorithm for calculating Ï€ to the nth digit, co-discoverer of the billionth, four billionth, 40th billionth, and quadrillionth digits of Ï€, and professor at Simon Fraser University. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ... The Dirichlet eta function can be defined as where ζ is Riemanns zeta function. ...

Applications

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. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Template:Otherusescccc A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ...

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: 1 + 2 + 3 + 4 + Â· Â· Â· is a divergent series. ...

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

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 In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, a divergent series is an infinite series that does not converge. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. ...

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

where the notation $(Re)$ indicates Ramanujan summation^[2].

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: In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ...

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

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. In mathematics and theoretical physics, zeta function regularization is a summability method assign finite values to superficially divergent sums. ... The mathematical term regularization has two main meanings, both associated with making a function more regular or smooth. ... In mathematics, a divergent series is an infinite series that does not converge. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In physics, the Casimir effect is a physical force exerted between separate objects, which is due to neither charge, gravity, nor the exchange of particles, but instead is due to resonance of all-pervasive energy fields in the intervening space between the objects. ...

Generalizations

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 There are a number of mathematical functions with the name zeta-function, named after the Greek letter Î¶. Of these, the most famous is the: Riemann zeta-function. ... In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...

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

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. In mathematics, a Dirichlet L-series, named in honour of Johann Peter Gustav Lejeune Dirichlet, is a function of the form Here Ï‡ is a Dirichlet character and s a complex variable with real part greater than 1. ... In mathematics, the Dedekind zeta-function is a Dirichlet series defined for any algebraic number field , and denoted where is a complex variable. ... There are a number of mathematical functions with the name zeta-function, named after the Greek letter Î¶. Of these, the most famous is the: Riemann zeta-function. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ...

The polylogarithm is given by The polylogarithm (also known as de JonquiÃ¨res function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ...

$Li_s(z) = sum_{k=1}^infty {z^k over k^s}$

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

The Lerch transcendent is given by In mathematics, the Lerch zeta function is a special function that generalizes the Hurwitz zeta function and the polylogarithm. ...

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

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 $L i s (e i θ)$

Zeta-functions in fiction

Neal Stephenson's 1999 novel Cryptonomicon mentions the zeta-function as a pseudo-random number source, a useful component in cipher design. Neal Town Stephenson (born October 31, 1959) is an American writer, known primarily for his science fiction works in the postcyberpunk genre with a penchant for explorations of society, mathematics, currency, and the history of science. ... 1999 (MCMXCIX) was a common year starting on Friday, and was designated the International Year of Older Persons by the United Nations. ... Cryptonomicon is a 1999 novel by Neal Stephenson. ... A pseudo-random number is a number belonging to a sequence which appears to be random, but can in fact be generated by a finite computation. ... This article is about algorithms for encryption and decryption. ...

The zeta-function is a major part of the plot of Thomas Pynchon's novel Against the Day (2006). Thomas Ruggles Pynchon, Jr. ... Against the Day is a novel by Thomas Pynchon to be published on November 21, 2006. ... For the Manfred Mann album, see 2006 (album). ...

The popular T.V. Show NUMB3RS had criminals who ransomed a child for a possible proof from a mathematician in order to steal interest rates from an encrypted website. NUMB3RS (Numbers) is an American television show that follows FBI Special Agent Don Eppes (Rob Morrow) and his mathematical genius brother, Charlie Eppes (David Krumholtz), who develops formulae to predict the actions of various criminals. ...

Notes

^ Ford, K. Vinogradov's integral and bounds for the Riemann zeta function, Proc. London Math. Soc. (3) 85 (2002), pp. 565-633
^ http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf

References

Bernhard Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse (1859). 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.

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.

Bernhard Riemann. ... This page is a candidate for speedy deletion. ... Helmut Hasse (pronounced HAHS uh) (25 August 1898- 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry (Hasse principle), and to local zeta functions. ... Edmund Taylor Whittaker (24 October 1873 - 24 March 1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. ... (George) Neville Watson (31 January 1886 - 2 February 1965) was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 â€“ December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... Edward Charles (Ted) Titchmarsh (born 1 June 1899 in Newbury died 18 January 1963 at Oxford) was a leading British mathematician. ... Jonathan M. Borwein (born 1951) was Shrum Professor of Science (1993-2003) and a Canada Research Chair in Information Technology (2001-08) at Simon Fraser University, and was founding Director of the Centre for Experimental and Constructive Mathematics. ... Richard E. Crandall is an American computer scientist who has made contributions to computational number theory. ...

External links

Riemann Zeta Function, in Wolfram Mathworld - an explanation with more mathematical approach
Tables of selected zeroes

Category: Zeta and L-functions

Results from FactBites:

Riemann zeta function - Wikipedia, the free encyclopedia (1557 words)
	In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers.
	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.
	Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory.

Riemann hypothesis - Wikipedia, the free encyclopedia (1847 words)
	The Riemann zeta function along the critical line is sometimes studied in terms of the Z function, whose real zeros correspond to the zeros of the zeta function on the critical line.
	Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof.
	The zeroes of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.

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Contents

Definition

Relationship to prime numbers