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

$\sum _{n}{\frac {a(n)}{n^{s}}}\,$

is equal to

$\prod _{p}P(p,s)\quad {\text{for }}\operatorname {Re} (s)>1.$

where the product is taken over prime numbers p, and P(p, s) is the sum

$\sum _{k=0}^{\infty }{\frac {a(p^{k})}{p^{ks}}}=1+{\frac {a(p)}{p^{s}}}+{\frac {a(p^{2})}{p^{2s}}}+{\frac {a(p^{3})}{p^{3s}}}+\cdots$

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(p^k) whenever n factors as the product of the powers p^k 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

$P(p,s)={\frac {1}{1-{\frac {a(p)}{p^{s}}}}},$

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

$\operatorname {Re} (s)>C,$

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 GL_m.

Examples

The following examples will use the notation $\mathbb {P}$ for the set of all primes, that is:

$\mathbb {P} =\{p\in \mathbb {N} \,|\,p{\text{ is prime}}\}.$

The Euler product attached to the Riemann zeta function ζ(s), also using the sum of the geometric series, is

${\begin{aligned}\prod _{p\,\in \,\mathbb {P} }\left({\frac {1}{1-{\frac {1}{p^{s}}}}}\right)&=\prod _{p\ \in \ \mathbb {P} }\left(\sum _{k=0}^{\infty }{\frac {1}{p^{ks}}}\right)\\&=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s).\end{aligned}}$

while for the Liouville function λ(n) = (−1)^ω(n), it is

$\prod _{p\,\in \,\mathbb {P} }\left({\frac {1}{1+{\frac {1}{p^{s}}}}}\right)=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}={\frac {\zeta (2s)}{\zeta (s)}}.$

Using their reciprocals, two Euler products for the Möbius function μ(n) are

$\prod _{p\,\in \,\mathbb {P} }\left(1-{\frac {1}{p^{s}}}\right)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}$

and

$\prod _{p\,\in \,\mathbb {P} }\left(1+{\frac {1}{p^{s}}}\right)=\sum _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}}.$

Taking the ratio of these two gives

$\prod _{p\,\in \,\mathbb {P} }\left({\frac {1+{\frac {1}{p^{s}}}}{1-{\frac {1}{p^{s}}}}}\right)=\prod _{p\,\in \,\mathbb {P} }\left({\frac {p^{s}+1}{p^{s}-1}}\right)={\frac {\zeta (s)^{2}}{\zeta (2s)}}.$

Since for even values of s the Riemann zeta function ζ(s) has an analytic expression in terms of a rational multiple of π^s, 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

${\begin{aligned}\prod _{p\,\in \,\mathbb {P} }\left({\frac {p^{2}+1}{p^{2}-1}}\right)&={\frac {5}{3}}\cdot {\frac {10}{8}}\cdot {\frac {26}{24}}\cdot {\frac {50}{48}}\cdot {\frac {122}{120}}\cdots &={\frac {\zeta (2)^{2}}{\zeta (4)}}&={\frac {5}{2}},\\[6pt]\prod _{p\,\in \,\mathbb {P} }\left({\frac {p^{4}+1}{p^{4}-1}}\right)&={\frac {17}{15}}\cdot {\frac {82}{80}}\cdot {\frac {626}{624}}\cdot {\frac {2402}{2400}}\cdots &={\frac {\zeta (4)^{2}}{\zeta (8)}}&={\frac {7}{6}},\end{aligned}}$

and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to

$\prod _{p\,\in \,\mathbb {P} }\left(1+{\frac {2}{p^{s}}}+{\frac {2}{p^{2s}}}+\cdots \right)=\sum _{n=1}^{\infty }{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta (s)^{2}}{\zeta (2s)}},$

where ω(n) counts the number of distinct prime factors of n, and 2^ω(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

$\prod _{p\,\in \,\mathbb {P} }{\frac {1}{1-{\frac {\chi (p)}{p^{s}}}}}=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.$

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

$\prod _{p\,\in \,\mathbb {P} }\left(x-{\frac {1}{p^{s}}}\right)\approx {\frac {1}{\operatorname {Li} _{s}(x)}}$

for s > 1 where Li_s(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

${\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots$

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):

${\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdots ,$

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:

$\prod _{p>2}\left(1-{\frac {1}{\left(p-1\right)^{2}}}\right)=0.660161...$

• The Landau–Ramanujan constant:

${\begin{aligned}{\frac {\pi }{4}}\prod _{p\equiv 1{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}&=0.764223...\\[6pt]{\frac {1}{\sqrt {2}}}\prod _{p\equiv 3{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}&=0.764223...\end{aligned}}$

• Murata's constant (sequence A065485 in the OEIS):

$\prod _{p}\left(1+{\frac {1}{\left(p-1\right)^{2}}}\right)=2.826419...$

• The strongly carefree constant ×ζ(2)² OEIS: A065472:

$\prod _{p}\left(1-{\frac {1}{\left(p+1\right)^{2}}}\right)=0.775883...$

• Artin's constant OEIS: A005596:

$\prod _{p}\left(1-{\frac {1}{p(p-1)}}\right)=0.373955...$

• Landau's totient constant OEIS: A082695:

$\prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)={\frac {315}{2\pi ^{4}}}\zeta (3)=1.943596...$

• The carefree constant ×ζ(2) OEIS: A065463:

$\prod _{p}\left(1-{\frac {1}{p(p+1)}}\right)=0.704442...$

and its reciprocal OEIS: A065489:

$\prod _{p}\left(1+{\frac {1}{p^{2}+p-1}}\right)=1.419562...$

• The Feller–Tornier constant OEIS: A065493:

${\frac {1}{2}}+{\frac {1}{2}}\prod _{p}\left(1-{\frac {2}{p^{2}}}\right)=0.661317...$

• The quadratic class number constant OEIS: A065465:

$\prod _{p}\left(1-{\frac {1}{p^{2}(p+1)}}\right)=0.881513...$

• The totient summatory constant OEIS: A065483:

$\prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=1.339784...$

• Sarnak's constant OEIS: A065476:

$\prod _{p>2}\left(1-{\frac {p+2}{p^{3}}}\right)=0.723648...$

• The carefree constant OEIS: A065464:

$\prod _{p}\left(1-{\frac {2p-1}{p^{3}}}\right)=0.428249...$

• The strongly carefree constant OEIS: A065473:

$\prod _{p}\left(1-{\frac {3p-2}{p^{3}}}\right)=0.286747...$

• Stephens' constant OEIS: A065478:

$\prod _{p}\left(1-{\frac {p}{p^{3}-1}}\right)=0.575959...$

• Barban's constant OEIS: A175640:

$\prod _{p}\left(1+{\frac {3p^{2}-1}{p(p+1)\left(p^{2}-1\right)}}\right)=2.596536...$

• Taniguchi's constant OEIS: A175639:

$\prod _{p}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)=0.678234...$

• The Heath-Brown and Moroz constant OEIS: A118228:

$\prod _{p}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)=0.0013176...$

References

• 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.