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 .
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
Many well known constants have Euler product expansions.
where each numerator is a prime number and each denominator is the nearest multiple of four.
Other Euler products for known constants include:
(with reciprocal) A065489:
- Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.
- 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, MR 0434929, Zbl 0335.10001 (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"