NationMaster - Encyclopedia: Euler product

In mathematics, an Euler product is an infinite product expansion, indexed by prime numbers p, of a Dirichlet series. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Euler. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... 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, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... 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, 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. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

In general, a Dirichlet series of the form

$sum_{n} a(n)n^{-s},$

where a(n) is a multiplicative function of n may be written as 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). ...

$prod_{p} P(p,s),$

where P(p,s) is the sum

$1+a(p)p^{-s} + a(p^2)p^{-2s} + cdots .$

In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that a(n) be multiplicative: this says exactly that a(n) is the product of the a(p^k) when n factorises as the product of the powers p^k of distinct primes p. In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

An important special case is that in which a(n) is totally multiplicative, so that P(p,s) is a geometric series. Then 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 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. ...

$P(p,s)=frac{1}{1-a(p)p^{-s}}$

as is the case for the Riemann zeta-function, where a(n) = 1), and more generally for Dirichlet characters. In number theory, a Dirichlet character is a function Ï‡ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that Ï‡(n) = Ï‡(n + k) for all n. ...

In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region In mathematics, a series is a sum of a sequence of terms. ...

Re(s) > C

that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.

In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here. The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GL_m. Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...

Examples of Euler products

The Euler product attached to the Riemann zeta function, using also the sum of the geometric series, is

$zeta(s) = sum_{n=1}^{infty}n^{-s} = prod_{p} Big(sum_{n=0}^{infty}p^{-ns}Big) = prod_{p} (1-p^{-s})^{-1}$ .

An Euler product for the Möbius function $μ(n)$ is The classical MÃ¶bius function is an important multiplicative function in number theory and combinatorics. ...

$frac{1}{zeta(s) }= prod_{p} (1-p^{-s})= sum_{n=1}^{infty}mu (n)n^{-s}$ .

Further products derived from the zeta function are

$frac{zeta(2s)}{zeta(s) }= prod_{p} (1+p^{-s})^{-1} = sum_{n=1}^{infty}lambda (n)n^{-s}$

where $λ(n) = ( - 1) Ω(n)$ is the Liouville function, and The Liouville function, denoted by Î»(n) and named after Joseph Liouville, is an important function in number theory. ...

$frac{zeta(s)}{zeta(2s) }= prod_{p} (1+p^{-s}) = sum_{n=1}^{infty} |mu(n)|n^{-s}$ .

Similarly

$frac{zeta(s)^2}{zeta(2s)} = prod_{p} Big(frac{1+p^{-s}}{1-p^{-s}}Big) = prod_{p} (1+2p^{-s}+2p^{-2s}+cdots) = sum_{n=1}^{infty}2^{omega(n)} n^{-s}$

where $ω(n)$ counts the number of distinct prime factors of n and $2 ω(n)$ the number of square-free divisors. In mathematics, a square-free integer is one divisible by no perfect square, except 1. ...

If $χ(n)$ is a Dirichlet character of conductor $N$ , so that $χ$ is totally multiplicative and $χ(n)$ only depends on n modulo N, and $χ(n) = 0$ if n is not coprime to N then Coprime - Wikipedia /**/ @import /skins-1. ...

$prod_{p} (1- chi(p) p^{-s})^{-1} = sum_{n=1}^{infty}chi(n)n^{-s}$ .

Here it is convenient to omit the primes p dividing the conductor N from the product.

References

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9 (Provides an introductory discussion of the Euler product in the context of classical number theory.)
G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 5th ed., Oxford (1979) ISBN 0-19-853171-0 (Chapter 17 gives further examples.))
Euler product on PlanetMath
Weisstein, Eric W., Euler Product at MathWorld.

Categories: Number theory | Zeta and L-functions PlanetMath is a free, collaborative, online mathematics encyclopedia. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

Euler product - definition of Euler product in Encyclopedia (292 words)
	The name arose from the case of the Riemann zeta function, where such a product representation was proved by Euler.
	This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.
	In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here.

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