Riemann zeta function

Riemann zeta function ζ(s) in the complex plane. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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 It also has applications in other areas such as physics, probability theory, and applied statistics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data.

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:

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

for values of s with real part greater than one, and then analytically continued to all complex s ≠ 1. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. This Dirichlet series converges for all real values of s greater than one. In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In Mathematics, the real numbers may be described informally in several different ways 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. Über die Anzahl der Primzahlen unter einer gegebenen Größe (Usual English translation On the Number of Primes Less Than a Given Magnitude) is a seminal Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z This article is about both real and complex analytic functions Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 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. In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values The celebrated Riemann hypothesis, formulated in the same paper of Riemann, is concerned with zeros of this analytically extended function. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved 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. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z

Euler product formula

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). In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 We will prove that the following formula holds \begin{align} \\\zeta(s & = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+ \cdots \\ & = \prod_{p} \frac{1}{1-p^{-s}} In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product In Number theory, an Euler product is an Infinite product expansion indexed by Prime numbers p, of a Dirichlet series. 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. We will prove that the following formula holds \begin{align} \\\zeta(s & = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+ \cdots \\ & = \prod_{p} \frac{1}{1-p^{-s}} In Mathematics, a geometric series is a series with a constant ratio between successive terms. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written Since the harmonic series, obtained when s = 1, diverges, Euler's formula implies that there are infinitely many primes. See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series

For s an integer number, the Euler product formula can be used to calculate the probability that s randomly selected integers are relatively prime. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than It turns out that this probability is indeed $\frac 1{\zeta(s)}.$

Various properties

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

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.

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

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

$\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty;\!$ this is the harmonic series. See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series

$\zeta(3/2) \approx 2.612;\!$ this is employed in calculating the critical temperature for a Bose–Einstein condensate in physics. A Bose–Einstein condensate (BEC is a State of matter of Bosons confined in an external Potential and cooled to Temperatures very near to

$\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.645;\!$ the demonstration of this equality is known as the Basel problem. The Basel problem is a famous problem in Number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735

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

$\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.202;\!$ this is called Apéry's constant. In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations

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

$\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.0823;\!$ Stefan–Boltzmann law and Wien approximation in physics. The Stefan–Boltzmann law, also known as Stefan's law, states that the total Energy radiated per unit surface Area of a Black body in unit Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of Physics used to describe the Spectrum

The functional equation

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 complex numbers s. In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values Here, Γ denotes the gamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function (This equation has to be interpreted analytically if any factors in the equation have a zero or pole. For instance, when s is 2, the right side has a simple zero in the sine factor and a simple pole in the Gamma factor, which cancel out and leave a nonzero finite value. Similarly, when s is 0, the right side has a simple zero in the sine factor and a simple pole in the zeta factor, which cancel out and leave a finite nonzero value. When s is 1, the right side has a simple pole in the Gamma factor that is not cancelled out by a zero in any other factor, which is consistent with the zeta-function on the left having a simple pole at 1. ) This formula, proved by Riemann in 1859, is used to construct the analytic continuation in the first place. Über die Anzahl der Primzahlen unter einer gegebenen Größe (Usual English translation On the Number of Primes Less Than a Given Magnitude) is a seminal (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. André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions ^[1]) At s = 1, the zeta-function has a simple pole with residue 1. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Complex analysis, the residue is a Complex number which describes the behavior of Line integrals of a Meromorphic function around a singularity 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 $\xi(t).\!$ ) 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)

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 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, 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 to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved 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. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved

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. John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} In Mathematics, the critical line theorem says that a positive proportion of the nontrivial zeros of the Riemann zeta function lie on the critical

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.

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.

Reciprocal

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. In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and For the rational functions defined on the complex numbers see Möbius transformation. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series. Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and

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 states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. In Mathematics, the universality of Zeta-functions is the remarkable property of the Riemann zeta-function and other similar functions such as the Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Since holomorphic functions are very general, this property is quite remarkable.

Representations

Mellin transform

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. In Mathematics, the Mellin transform is an Integral transform that may be regarded as the multiplicative version of the Two-sided Laplace transform 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. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial 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

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

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. In Mathematics, the prime-counting function is the function counting the number of Prime numbers less than or equal to some Real number x Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion. In Mathematics, the classic Möbius inversion formula was introduced into Number theory during the 19th century by August Ferdinand Möbius.

Also, from the above (specifically, the second equation in this section), we can write the zeta function in the commonly seen form:

$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \left(\frac{x^{s-1}}{\exp(x)-1}\right) \ dx.\!$

Laurent series

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 It can therefore be expanded as a Laurent series about s = 1; the series development then is

$\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n \; (s-1)^n.$

The constants γ_n here are called the Stieltjes constants and can be defined by the limit

$\gamma_n = \lim_{m \rightarrow \infty}{\left(\left(\sum_{k = 1}^m \frac{(\ln k)^n}{k}\right) - \frac{(\ln m)^{n+1}}{n+1}\right)}.$

The constant term γ₀ is the Euler-Mascheroni constant. In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms In Mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function: In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually

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 (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling

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 Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling

Hadamard product

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

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

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler-Mascheroni constant. In Mathematics, the Weierstrass factorization theorem in Complex analysis, named after Karl Weierstrass, asserts that Entire functions can be Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually A simpler infinite product expansion is

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

This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, . In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product . . due to the gamma function term in the denominator, and the non-trivial zeros at s=ρ.

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:

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

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). Konrad Hermann Theodor Knopp ( 22 July 1882, Berlin, Germany – 20 April 1957, Annecy, France) was Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic Year 1930 ( MCMXXX) was a Common year starting on Wednesday (link will display 1930 calendar of the Gregorian calendar.

Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. Peter Benjamin Borwein ( St Andrews, Scotland, 1953 is a Canadian Mathematician, co-developer of an algorithm for calculating π The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function. In Mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of Orthogonal polynomials which are related to In Mathematics, in the area of Analytic number theory, the Dirichlet eta function can be defined as \eta(s = \left(1-2^{1-s}\right

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 Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one 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:

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

The sum S appears to take the form of $\zeta(-1).\!$ However, −1 lies outside of the domain for which the Dirichlet series for the zeta-function converges. The sum of all Natural numbers 1 + 2 + 3 + 4 + · · ·, also written \sum_{n=1}^{\infin} n^1 is a Divergent series; the In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined 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. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite 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 Mathematics, the Euler–Maclaurin formula provides a powerful connection between Integrals (see Calculus) and sums 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 Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. In Mathematics and Theoretical physics, zeta-function regularization is a type of regularization or Summability method that assigns finite values In Physics, especially Quantum field theory, regularization is a method of dealing with infinite divergent and non-sensical expressions by introducing an auxiliary In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums In quantum field theory (QFT the forces between particles are mediated by other particles In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect. In Physics, the Casimir effect and the Casimir-Polder force are physical forces arising from a quantized field.

Generalizations

There are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. A zeta function is a function which is composed of an infinite sum of powers that is which may be written as a Dirichlet series: \zeta(s = \sum_{k=1}^{\infty}f(k^s 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. In Mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many Zeta functions It is formally defined for complex arguments In mathematics a Dirichlet L -series, named in honour of Johann Peter Gustav Lejeune Dirichlet, is a function of the form L(s\chi = \sum_{n=1}^\infty In Mathematics, the Dedekind zeta-function is a Dirichlet series defined for any Algebraic number field K and denoted \zeta_K For other related functions see the articles Zeta function and L-function. A zeta function is a function which is composed of an infinite sum of powers that is which may be written as a Dirichlet series: \zeta(s = \sum_{k=1}^{\infty}f(k^s The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory.

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 polylogarithm (also known as de Jonquière's function) is a Special function Li s ( z) that is defined by the sum

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). In Mathematics, the Lerch zeta-function, sometimes called the Hurwitz-Lerch zeta-function, is a Special function that generalizes the Hurwitz

The Clausen function $Cl_{s} ( \theta )\!$ that can be chosen as the real or imaginary part of $\mathrm{Li}_{s} (e^{i\theta})\!$

The multiple zeta functions are defined by

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

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.

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 Year 1999 ( MCMXCIX) was a Common year starting on Friday (link will display full 1999 Gregorian calendar) Cryptonomicon is a 1999 novel by Neal Stephenson. It concurrently follows the exploits of World War II -era Cryptographers affiliated with A pseudorandom process is a process that appears random but is not In Cryptography, a cipher (or cypher) is an Algorithm for performing Encryption and Decryption &mdash a series of well-defined steps

The zeta-function is a major part of the plot of Thomas Pynchon's 2006 novel Against the Day. Thomas Ruggles Pynchon Jr (born May 8 1937 is an American writer based in New York City, noted for his dense and complex works of Fiction. Against the Day is a Novel by Thomas Pynchon. The Narrative takes place between the 1893 Chicago World's Fair and the time immediately

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. NUMB3RS (pronounced Numbers) is an American Television show produced by brothers Ridley and Tony Scott. The proof would be used to steal interest rates from an encrypted website.

Strip number 113 of the webcomic XKCD has a person describing themselves as being like the Riemann zeta function. xkcd is a Webcomic created by Randall Munroe, a former contractor for NASA. ^[5]

Notes

^ "Euler and the Zeta Function" by Raymond Ayoub, American Mathematical Monthly, v. The Riemann hypothesis is one of the most important Conjectures in Mathematics. In Mathematics, the Riemann-Siegel theta function is defined in terms of the Gamma function as \theta(t = \arg \left( \Gamma\left(\frac{2it+1}{4}\right\right The American Mathematical Monthly ( is a mathematical journal founded by Benjamin Finkel in 1894. 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. doi:10.1088/0305-4470/39/45/008. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Eric Delabaere (December 3, 2001), Ramanujan’s Summation, Universit´e d’Angers (France), <http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf>. Events 1800 - War of the Second Coalition: Battle of Hohenlinden, French Year 2001 ( MMI) was a Common year starting on Monday according to the Gregorian calendar. Retrieved on 8 May 2008
^ xkcd - A webcomic of romance, sarcasm, math, and language - By Randall Munroe

References

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. Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of
Helmut Hasse, Ein Summierungsverfahren für die Riemannsche ζ-Reihe, (1930) Math. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic Z. 32 pp 458-464. (Globally convergent series expression. )
E. T. Whittaker and G. N. Watson (1927). Edmund Taylor Whittaker ( 24 October[[ 873]] - 24 March[[ 956]] was a mathematician who contributed widely to Applied mathematics, Mathematical physics (George Neville Watson ( 31 January 1886 – 2 February 1965) was an English mathematician a noted master in the application of Complex 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). Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 Divergent Series. Clarendon Press, Oxford.
A. Ivic (1985). The Riemann Zeta Function. John Wiley & Sons. ISBN 0-471-80634-X.
E. C. Titchmarsh (1986). Edward Charles ("Ted" Titchmarsh (born 1 June 1899 in Newbury died 18 January, 1963 at Oxford) was a leading The Theory of the Riemann Zeta Function, Second revised (Heath-Brown) edition. Oxford University Press.
Jonathan Borwein, David M. Jonathan Michael Borwein (born 1951 is a Canadian mathematician noted for his prolific and creative work throughout the international mathematical community Bradley, Richard Crandall (2000). Richard E Crandall is an American Computer scientist and physicist who has made contributions to Computational number theory, most notably the development "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. doi:10.1016/S0377-0427(02)00358-8. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

Djurdje Cvijović and Jacek Klinowski (1997). "Continued-fraction expansions for the Riemann zeta function and polylogarithms". Proc. Amer. Math. Soc. 125: pp. 2543–2550. doi:10.1090/S0002-9939-97-04102-6. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

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.

External links

Riemann Zeta Function, in Wolfram Mathworld - an explanation with a 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.
X-Ray of the Zeta Function Visually-oriented investigation of where zeta is real or purely imaginary. (PDF)
Formulas and identities for the Riemann Zeta function functions. wolfram. com
Riemann Zeta Function and Other Sums of Reciprocal Powers, section 23. 2 of Abramowitz and Stegun

Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U

Dictionary