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.
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
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
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 is 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. In his notebooks, Ramanujan generalized the Euler product for the zeta function as
for where is the polylogarithm. For the product above is just
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:
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:
- 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"