Euler product
expansions of functions into infinite products
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
∑
n
=
1
∞
a
(
n
)
n
s
{\displaystyle \sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}}
is equal to
∏
p
∈
P
P
(
p
,
s
)
for
Re
(
s
)
>
1.
{\displaystyle \prod _{p\in \mathbb {P} }P(p,s)\quad {\text{for }}\operatorname {Re} (s)>1.}
where the product is taken over prime numbers p, and P(p, s) is the sum
∑
k
=
0
∞
a
(
p
k
)
p
k
s
=
1
+
a
(
p
)
p
s
+
a
(
p
2
)
p
2
s
+
a
(
p
3
)
p
3
s
+
⋯
{\displaystyle \sum _{k=0}^{\infty }{\frac {a(p^{k})}{p^{ks}}}=1+{\frac {a(p)}{p^{s}}}+{\frac {a(p^{2})}{p^{2s}}}+{\frac {a(p^{3})}{p^{3s}}}+\cdots }
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
P
(
p
,
s
)
=
1
1
−
a
(
p
)
p
s
,
{\displaystyle P(p,s)={\frac {1}{1-{\frac {a(p)}{p^{s}}}}},}
as is the case for the Riemann zeta function, where a(n) = 1, 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
Re
(
s
)
>
C
,
{\displaystyle \operatorname {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.
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