Euler product

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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.

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

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

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 factors as the product of the powers p^k of distinct primes p.

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

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

as is the case for the Riemann zeta-function, where a(n) = 1), and more generally for Dirichlet characters.

In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region

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.

[edit] 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

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

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

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

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.

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

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

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

Ramanujan in his notebooks tried to generalize the Euler product for Zeta function in the form:

$\prod_{p} (x-p^{-s})\approx \frac{1}{Li_{s} (x)}$ s > 1

where Li_s (x) is the Polylogarithm for x=1 the product above is just $1 / ζ(s)$

searching for a way to obtain prime powers as roots of a certain function f(x, s).

[edit] References

G. Polya, Induction and Analogy in Mathematics Volume 1 (1954) Princeton University Press L.C. Card 53-6388 (A very accessible English translation of Euler's memoir regarding this "Most Extraordinary Law of the Numbers" appears starting on page 91)
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.))'
'Ramanujan lost notebook'[[1]]
Euler product on PlanetMath
Eric W. Weisstein, Euler Product at MathWorld.