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. 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} \mid 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.
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. Dividing the Euler product for ζ ( 2 ) = π 2 6 {\displaystyle \zeta (2)={\frac {\pi ^{2}}{6}}} by the previous product, known as Basel problem, one finds that
π 2 = ( ∏ p ≡ 1 ( mod 4 ) p p + 1 ) ( ∏ p ≡ 3 ( mod 4 ) p p − 1 ) . {\displaystyle {\frac {\pi }{2}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p+1}}\right)\left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p-1}}\right).}
Taking the ratio of the previous two products gives
2 = ( ∏ p ≡ 1 ( mod 4 ) p − 1 p + 1 ) ( ∏ p ≡ 3 ( mod 4 ) p + 1 p − 1 ) . {\displaystyle 2=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p-1}{p+1}}\right)\left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p+1}{p-1}}\right).}
The infinite products must be taken in order of increasing primes. 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\ldots }
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\ldots \\[6pt]{\frac {1}{\sqrt {2}}}\prod _{p\equiv 3{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}&=0.764223\ldots \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\ldots }
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\ldots }
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\ldots }
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\ldots }
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\ldots }
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\ldots }
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\ldots }
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\ldots }
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\ldots }
Sarnak's constant OEIS: A065476:
∏ p > 2 ( 1 −