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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} \,|\,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 4.¹

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

• 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"

External links

• This article incorporates material from Euler product on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• Stepanov, S.A. (2001) [1994], "Euler product", Encyclopedia of Mathematics, EMS Press
• Weisstein, Eric W. "Euler Product". MathWorld.
• Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived from the original on 12 June 2006.

References

1. Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267. 9781848165267 ↩