In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.

Definition

In general, if a is a bounded multiplicative function, then the Dirichlet series is equal to where the product is taken over prime numbers p, and P(p, s) 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 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 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 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 following examples will use the notation \mathbb{P} for the set of all primes, that is: The Euler product attached to the Riemann zeta function ζ(s) , also using the sum of the geometric series, is while for the Liouville function , it is Using their reciprocals, two Euler products for the Möbius function μ(n) are and Taking the ratio of these two gives Since for even values of 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 , , and , then and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to where ω(n) counts the number of distinct prime factors of n, and 2ω(n) is 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 mod 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. In his notebooks, Ramanujan generalized the Euler product for the zeta function as for s > 1 where Lis(x) is the polylogarithm. For the product above is just 1⁄ζ(s) .

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 (fractions where numerator and denominator differ by 1): where each numerator is a prime number and each denominator is the nearest multiple of 4. Other Euler products for known constants include: ×ζ(2)2 ×ζ(2)