# 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 is a multiplicative function, then the Dirichlet series

is equal to

where the product is taken over prime numbers , and is the sum

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 be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes .

An important special case is that in which is totally multiplicative, so that is a geometric series. Then

as is the case for the Riemann zeta-function, where , 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 GL_{m}.

## Examples

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

while for the Liouville function , it is,

Using their reciprocals, two Euler products for the Möbius function are,

and,

and taking the ratio of these two gives,

Since for even * s* the Riemann zeta function has an analytic expression in terms of a

*rational*multiple of , then for even exponents, this infinite product evaluates to a rational number. For example, since , , and , then,

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

where counts the number of distinct prime factors of *n* and the number of square-free divisors.

If is a Dirichlet character of *conductor* , so that is totally multiplicative and only depends on *n* modulo *N*, and if *n* is not coprime to *N* then,

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:

for where is the polylogarithm. For the product above is just

## Notable constants

Many well known constants have Euler product expansions.

can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios

where each numerator is a prime number and each denominator is the nearest multiple of four.^{[1]}

Other Euler products for known constants include:

Murata's constant (sequence A065485 in OEIS):

Strongly carefree constant A065472:

Landau's totient constant A082695:

(with reciprocal) A065489:

Feller-Tornier constant A065493:

Quadratic class number constant A065465:

Totient summatory constant A065483:

Strongly carefree constant A065473:

Heath-Brown and Moroz constant A118228:

## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.

## 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)* - Template:Apostol IANT
*(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

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