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.


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.


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.


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 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.

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.[1]

Other Euler products for known constants include:

Twin prime constant:

Landau-Ramanujan constant:

Murata's constant (sequence A065485 in the OEIS):

Strongly carefree constant A065472:

Artin's constant A005596:

Landau's totient constant A082695:

Carefree constant A065463:

(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:


  1. Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.


External links

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