Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.

Definition

In general, if a is a multiplicative function, then the Dirichlet series

\[ \sum_{n} a(n)n^{-s}\, \]

is equal to

\[ \prod_{p} P(p, s)\, \]

where the product is taken over prime numbers p, and 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(pk) whenever n factors as the product of the powers pk 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.
Convergence

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 GLm.
Examples

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

\[ \prod_{p} (1-p^{-s})^{-1} = \prod_{p} \Big(\sum_{n=0}^{\infty}p^{-ns}\Big) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} = \zeta(s) . \]

while for the Liouville function λ(n) = ( − 1)Ω(n), it is,

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

Using their reciprocals, two Euler products for the Möbius function μ(n) are,

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

and,

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

and taking the ratio of these two gives,

\[ \prod_{p} \Big(\frac{1+p^{-s}}{1-p^{-s}}\Big) = \prod_{p} \Big(\frac{p^{s}+1}{p^{s}-1}\Big) = \frac{\zeta(s)^2}{\zeta(2s)} \]

Since for even s the Riemann zeta function ζ(s) has an analytic expression in terms of a rational multiple of πs, then for even exponents, this infinite product evaluates to a rational number. For example, since ζ(2) = π² / 6, ζ(4) = π⁴ / 90, and ζ(8) = π⁸ / 9450, then,

\[ \prod_{p} \Big(\frac{p^{2}+1}{p^{2}-1}\Big) = \frac{5}{2} \]

\[ \prod_{p} \Big(\frac{p^{4}+1}{p^{4}-1}\Big) = \frac{7}{6} \]

and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to,

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

where ω(n) counts the number of distinct prime factors of n and 2ω(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}{\operatorname{Li}_{s} (x)}

for s > 1 where \operatorname{Li}_s(x) is the polylogarithm. For x = 1 the product above is just 1 / ζ(s).
Notable constants

Many well known constants have Euler product expansions.

Twin prime constant:

\[ \prod_{p>2} \Big(1 - \frac{1}{(p-1)^2}\Big) = 0.660161... \]

Landau-Ramanujan constant:

\[ \frac{\pi}{4} \prod_{p = 1\,\text{mod}\,4} \Big(1 - \frac{1}{p^2}\Big)^{1/2} = 0.764223... \]

\[ \frac{1}{\sqrt{2}} \prod_{p = 3\,\text{mod}\,4} \Big(1 - \frac{1}{p^2}\Big)^{-1/2} = 0.764223... \]

Murata's constant (sequence A065485 in OEIS):

\[ \prod_{p} \Big(1 + \frac{1}{(p-1)^2}\Big) = 2.826419... \]

Strongly carefree constant \times\zeta(2)^2 OEIS A065472:

\[ \prod_{p} \Big(1 - \frac{1}{(p+1)^2}\Big) = 0.775883... \]

Artin's constant OEIS A005596:

\[ \prod_{p} \Big(1 - \frac{1}{p(p-1)}\Big) = 0.373955... \]

Landau's totient constant OEIS A082695:

\[ \prod_{p} \Big(1 + \frac{1}{p(p-1)}\Big) = \frac{315}{2\pi^4}\zeta(3) = 1.943596... \]

Carefree constant \times\zeta(2) OEIS A065463:

\[ \prod_{p} \Big(1 - \frac{1}{p(p+1)}\Big) = 0.704442... \]

(with reciprocal) OEIS A065489:

\[ \prod_{p} \Big(1 + \frac{1}{p^2+p-1}\Big) = 1.419562... \]

Feller-Tornier constant OEIS A065493:

\[ \frac{1}{2}+\frac{1}{2} \prod_{p} \Big(1 - \frac{2}{p^2}\Big) = 0.661317... \]

Quadratic class number constant OEIS A065465:

\[ \prod_{p} \Big(1 - \frac{1}{p^2(p+1)}\Big) = 0.881513... \]

Totient summatory constant OEIS A065483:

\[ \prod_{p} \Big(1 + \frac{1}{p^2(p-1)}\Big) = 1.339784... \]

Sarnak's constant OEIS A065476:

\[ \prod_{p>2} \Big(1 - \frac{p+2}{p^3}\Big) = 0.723648... \]

Carefree constant OEIS A065464:

\[ \prod_{p} \Big(1 - \frac{2p-1}{p^3}\Big) = 0.428249... \]

Strongly carefree constant OEIS A065473:

\[ \prod_{p} \Big(1 - \frac{3p-2}{p^3}\Big) = 0.286747... \]

Stephens' constant OEIS A065478:

\[ \prod_{p} \Big(1 - \frac{p}{p^3-1}\Big) = 0.575959... \]

Barban's constant OEIS A175640:

\[ \prod_{p} \Big(1 + \frac{3p^2-1}{p(p+1)(p^2-1)}\Big) = 2.596536... \]

Taniguchi's constant OEIS A175639:

\[ \prod_{p} \Big(1 - \frac{3}{p^3}+\frac{2}{p^4}+\frac{1}{p^5}-\frac{1}{p^6}\Big) = 0.678234... \]

Heath-Brown and Moroz constant OEIS A118228:

\[ \prod_{p} \Big(1 - \frac{1}{p}\Big)^7 \Big(1 + \frac{7p+1}{p^2}\Big) = 0.0013176... \]

References

G. Polya, Induction and Analogy in Mathematics Volume 1 Princeton University Press (1954) 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)
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929 (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.)
George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part I, Springer (2005), ISBN 0-387-25529-X
G. Niklasch, Some number theoretical constants: 1000-digit values"

External links

Euler product on PlanetMath
Weisstein, Eric W., "Euler Product" from MathWorld.
Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived from the original on 12 Jun 2006.