- Source: Euler product
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
a
(
n
)
n
s
{\displaystyle \sum _{n}{\frac {a(n)}{n^{s}}}\,}
is equal to
∏
p
P
(
p
,
s
)
for
Re
(
s
)
>
1.
{\displaystyle \prod _{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. 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
P
{\displaystyle \mathbb {P} }
for the set of all primes, that is:
P
=
{
p
∈
N
|
p
is prime
}
.
{\displaystyle \mathbb {P} =\{p\in \mathbb {N} \,|\,p{\text{ is prime}}\}.}
The Euler product attached to the Riemann zeta function ζ(s), also using the sum of the geometric series, is
∏
p
∈
P
(
1
1
−
1
p
s
)
=
∏
p
∈
P
(
∑
k
=
0
∞
1
p
k
s
)
=
∑
n
=
1
∞
1
n
s
=
ζ
(
s
)
.
{\displaystyle {\begin{aligned}\prod _{p\,\in \,\mathbb {P} }\left({\frac {1}{1-{\frac {1}{p^{s}}}}}\right)&=\prod _{p\ \in \ \mathbb {P} }\left(\sum _{k=0}^{\infty }{\frac {1}{p^{ks}}}\right)\\&=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s).\end{aligned}}}
while for the Liouville function λ(n) = (−1)ω(n), it is
∏
p
∈
P
(
1
1
+
1
p
s
)
=
∑
n
=
1
∞
λ
(
n
)
n
s
=
ζ
(
2
s
)
ζ
(
s
)
.
{\displaystyle \prod _{p\,\in \,\mathbb {P} }\left({\frac {1}{1+{\frac {1}{p^{s}}}}}\right)=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}={\frac {\zeta (2s)}{\zeta (s)}}.}
Using their reciprocals, two Euler products for the Möbius function μ(n) are
∏
p
∈
P
(
1
−
1
p
s
)
=
∑
n
=
1
∞
μ
(
n
)
n
s
=
1
ζ
(
s
)
{\displaystyle \prod _{p\,\in \,\mathbb {P} }\left(1-{\frac {1}{p^{s}}}\right)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}}
and
∏
p
∈
P
(
1
+
1
p
s
)
=
∑
n
=
1
∞
|
μ
(
n
)
|
n
s
=
ζ
(
s
)
ζ
(
2
s
)
.
{\displaystyle \prod _{p\,\in \,\mathbb {P} }\left(1+{\frac {1}{p^{s}}}\right)=\sum _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}}.}
Taking the ratio of these two gives
∏
p
∈
P
(
1
+
1
p
s
1
−
1
p
s
)
=
∏
p
∈
P
(
p
s
+
1
p
s
−
1
)
=
ζ
(
s
)
2
ζ
(
2
s
)
.
{\displaystyle \prod _{p\,\in \,\mathbb {P} }\left({\frac {1+{\frac {1}{p^{s}}}}{1-{\frac {1}{p^{s}}}}}\right)=\prod _{p\,\in \,\mathbb {P} }\left({\frac {p^{s}+1}{p^{s}-1}}\right)={\frac {\zeta (s)^{2}}{\zeta (2s)}}.}
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 ζ(2) = π2/6, ζ(4) = π4/90, and ζ(8) = π8/9450, then
∏
p
∈
P
(
p
2
+
1
p
2
−
1
)
=
5
3
⋅
10
8
⋅
26
24
⋅
50
48
⋅
122
120
⋯
=
ζ
(
2
)
2
ζ
(
4
)
=
5
2
,
∏
p
∈
P
(
p
4
+
1
p
4
−
1
)
=
17
15
⋅
82
80
⋅
626
624
⋅
2402
2400
⋯
=
ζ
(
4
)
2
ζ
(
8
)
=
7
6
,
{\displaystyle {\begin{aligned}\prod _{p\,\in \,\mathbb {P} }\left({\frac {p^{2}+1}{p^{2}-1}}\right)&={\frac {5}{3}}\cdot {\frac {10}{8}}\cdot {\frac {26}{24}}\cdot {\frac {50}{48}}\cdot {\frac {122}{120}}\cdots &={\frac {\zeta (2)^{2}}{\zeta (4)}}&={\frac {5}{2}},\\[6pt]\prod _{p\,\in \,\mathbb {P} }\left({\frac {p^{4}+1}{p^{4}-1}}\right)&={\frac {17}{15}}\cdot {\frac {82}{80}}\cdot {\frac {626}{624}}\cdot {\frac {2402}{2400}}\cdots &={\frac {\zeta (4)^{2}}{\zeta (8)}}&={\frac {7}{6}},\end{aligned}}}
and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to
∏
p
∈
P
(
1
+
2
p
s
+
2
p
2
s
+
⋯
)
=
∑
n
=
1
∞
2
ω
(
n
)
n
s
=
ζ
(
s
)
2
ζ
(
2
s
)
,
{\displaystyle \prod _{p\,\in \,\mathbb {P} }\left(1+{\frac {2}{p^{s}}}+{\frac {2}{p^{2s}}}+\cdots \right)=\sum _{n=1}^{\infty }{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta (s)^{2}}{\zeta (2s)}},}
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 χ(n) = 0 if n is not coprime to N, then
∏
p
∈
P
1
1
−
χ
(
p
)
p
s
=
∑
n
=
1
∞
χ
(
n
)
n
s
.
{\displaystyle \prod _{p\,\in \,\mathbb {P} }{\frac {1}{1-{\frac {\chi (p)}{p^{s}}}}}=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.}
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
∏
p
∈
P
(
x
−
1
p
s
)
≈
1
Li
s
(
x
)
{\displaystyle \prod _{p\,\in \,\mathbb {P} }\left(x-{\frac {1}{p^{s}}}\right)\approx {\frac {1}{\operatorname {Li} _{s}(x)}}}
for s > 1 where Lis(x) is the polylogarithm. For x = 1 the product above is just 1/ζ(s).
Notable constants
Many well known constants have Euler product expansions.
The Leibniz formula for π
π
4
=
∑
n
=
0
∞
(
−
1
)
n
2
n
+
1
=
1
−
1
3
+
1
5
−
1
7
+
⋯
{\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }
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):
π
4
=
(
∏
p
≡
1
(
mod
4
)
p
p
−
1
)
(
∏
p
≡
3
(
mod
4
)
p
p
+
1
)
=
3
4
⋅
5
4
⋅
7
8
⋅
11
12
⋅
13
12
⋯
,
{\displaystyle {\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdots ,}
where each numerator is a prime number and each denominator is the nearest multiple of 4.
Other Euler products for known constants include:
The Hardy–Littlewood twin prime constant:
∏
p
>
2
(
1
−
1
(
p
−
1
)
2
)
=
0.660161...
{\displaystyle \prod _{p>2}\left(1-{\frac {1}{\left(p-1\right)^{2}}}\right)=0.660161...}
The Landau–Ramanujan constant:
π
4
∏
p
≡
1
(
mod
4
)
(
1
−
1
p
2
)
1
2
=
0.764223...
1
2
∏
p
≡
3
(
mod
4
)
(
1
−
1
p
2
)
−
1
2
=
0.764223...
{\displaystyle {\begin{aligned}{\frac {\pi }{4}}\prod _{p\equiv 1{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}&=0.764223...\\[6pt]{\frac {1}{\sqrt {2}}}\prod _{p\equiv 3{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}&=0.764223...\end{aligned}}}
Murata's constant (sequence A065485 in the OEIS):
∏
p
(
1
+
1
(
p
−
1
)
2
)
=
2.826419...
{\displaystyle \prod _{p}\left(1+{\frac {1}{\left(p-1\right)^{2}}}\right)=2.826419...}
The strongly carefree constant ×ζ(2)2 OEIS: A065472:
∏
p
(
1
−
1
(
p
+
1
)
2
)
=
0.775883...
{\displaystyle \prod _{p}\left(1-{\frac {1}{\left(p+1\right)^{2}}}\right)=0.775883...}
Artin's constant OEIS: A005596:
∏
p
(
1
−
1
p
(
p
−
1
)
)
=
0.373955...
{\displaystyle \prod _{p}\left(1-{\frac {1}{p(p-1)}}\right)=0.373955...}
Landau's totient constant OEIS: A082695:
∏
p
(
1
+
1
p
(
p
−
1
)
)
=
315
2
π
4
ζ
(
3
)
=
1.943596...
{\displaystyle \prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)={\frac {315}{2\pi ^{4}}}\zeta (3)=1.943596...}
The carefree constant ×ζ(2) OEIS: A065463:
∏
p
(
1
−
1
p
(
p
+
1
)
)
=
0.704442...
{\displaystyle \prod _{p}\left(1-{\frac {1}{p(p+1)}}\right)=0.704442...}
and its reciprocal OEIS: A065489:
∏
p
(
1
+
1
p
2
+
p
−
1
)
=
1.419562...
{\displaystyle \prod _{p}\left(1+{\frac {1}{p^{2}+p-1}}\right)=1.419562...}
The Feller–Tornier constant OEIS: A065493:
1
2
+
1
2
∏
p
(
1
−
2
p
2
)
=
0.661317...
{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\prod _{p}\left(1-{\frac {2}{p^{2}}}\right)=0.661317...}
The quadratic class number constant OEIS: A065465:
∏
p
(
1
−
1
p
2
(
p
+
1
)
)
=
0.881513...
{\displaystyle \prod _{p}\left(1-{\frac {1}{p^{2}(p+1)}}\right)=0.881513...}
The totient summatory constant OEIS: A065483:
∏
p
(
1
+
1
p
2
(
p
−
1
)
)
=
1.339784...
{\displaystyle \prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=1.339784...}
Sarnak's constant OEIS: A065476:
∏
p
>
2
(
1
−
p
+
2
p
3
)
=
0.723648...
{\displaystyle \prod _{p>2}\left(1-{\frac {p+2}{p^{3}}}\right)=0.723648...}
The carefree constant OEIS: A065464:
∏
p
(
1
−
2
p
−
1
p
3
)
=
0.428249...
{\displaystyle \prod _{p}\left(1-{\frac {2p-1}{p^{3}}}\right)=0.428249...}
The strongly carefree constant OEIS: A065473:
∏
p
(
1
−
3
p
−
2
p
3
)
=
0.286747...
{\displaystyle \prod _{p}\left(1-{\frac {3p-2}{p^{3}}}\right)=0.286747...}
Stephens' constant OEIS: A065478:
∏
p
(
1
−
p
p
3
−
1
)
=
0.575959...
{\displaystyle \prod _{p}\left(1-{\frac {p}{p^{3}-1}}\right)=0.575959...}
Barban's constant OEIS: A175640:
∏
p
(
1
+
3
p
2
−
1
p
(
p
+
1
)
(
p
2
−
1
)
)
=
2.596536...
{\displaystyle \prod _{p}\left(1+{\frac {3p^{2}-1}{p(p+1)\left(p^{2}-1\right)}}\right)=2.596536...}
Taniguchi's constant OEIS: A175639:
∏
p
(
1
−
3
p
3
+
2
p
4
+
1
p
5
−
1
p
6
)
=
0.678234...
{\displaystyle \prod _{p}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)=0.678234...}
The Heath-Brown and Moroz constant OEIS: A118228:
∏
p
(
1
−
1
p
)
7
(
1
+
7
p
+
1
p
2
)
=
0.0013176...
{\displaystyle \prod _{p}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)=0.0013176...}
Notes
References
External links
Kata Kunci Pencarian:
- Usaha (fisika)
- Kalsium oksalat
- Daftar bilangan prima
- Daftar tetapan matematis
- Teorema Fubini
- Roger Cotes
- Teknik struktur
- Deret harmonik (matematika)
- Bilangan riil negatif
- Teorema dasar aljabar
- Euler product
- Proof of the Euler product formula for the Riemann zeta function
- Riemann zeta function
- List of topics named after Leonhard Euler
- Euler characteristic
- Euler's totient function
- Leibniz formula for π
- Dirichlet L-function
- Dedekind zeta function
- Euler system
