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.
Contents
Definition
In general, if
is equal to
where the product is taken over prime numbers
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
An important special case is that in which
as is the case for the Riemann zeta-function, where
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) > Cthat 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
while for the Liouville function
Using their reciprocals, two Euler products for the Möbius function
and,
and taking the ratio of these two gives,
Since for even s the Riemann zeta function
and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to,
where
If
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
Notable constants
Many well known constants have Euler product expansions.
The Leibniz formula for π,
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.
Other Euler products for known constants include:
Hardy–Littlewood's twin prime constant:
Landau-Ramanujan constant:
Murata's constant (sequence A065485 in the OEIS):
Strongly carefree constant
Artin's constant A005596:
Landau's totient constant A082695:
Carefree constant
(with reciprocal) A065489:
Feller-Tornier constant A065493:
Quadratic class number constant A065465:
Totient summatory constant A065483:
Sarnak's constant A065476:
Carefree constant A065464:
Strongly carefree constant A065473:
Stephens' constant A065478:
Barban's constant A175640:
Taniguchi's constant A175639:
Heath-Brown and Moroz constant A118228: