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Concept
Euler product
Summary
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 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 a(n) = 1, 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 following examples will use the notation 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 λ(n) = (−1)ω(n), 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.
Official source
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