Euler product

In general, if ${\displaystyle a}$ is a multiplicative function, then the Dirichlet series

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

is equal to

${\displaystyle \prod _{p}P(p,s)\,}$

where the product is taken over prime numbers ${\displaystyle p}$, and ${\displaystyle P(p,s)}$ is the sum

${\displaystyle 1+a(p)p^{-s}+a(p^{2})p^{-2s}+\cdots .}$

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 ${\displaystyle a(n)}$ be multiplicative: this says exactly that ${\displaystyle a(n)}$ is the product of the ${\displaystyle a(p^{k})}$ whenever ${\displaystyle n}$ factors as the product of the powers ${\displaystyle p^{k}}$ of distinct primes ${\displaystyle p}$.

An important special case is that in which ${\displaystyle a(n)}$ is totally multiplicative, so that ${\displaystyle P(p,s)}$ is a geometric series. Then

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

as is the case for the Riemann zeta-function, where ${\displaystyle a(n)=1}$, and more generally for Dirichlet characters.

In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region

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

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 GL_m.

The Euler product attached to the Riemann zeta function ${\displaystyle \zeta (s),}$ using also the sum of the geometric series, is

${\displaystyle \prod _{p}(1-p^{-s})^{-1}=\prod _{p}{\Big (}\sum _{n=0}^{\infty }p^{-ns}{\Big )}=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s).}$

while for the Liouville function ${\displaystyle \lambda (n)=(-1)^{\Omega (n)},}$ it is

${\displaystyle \prod _{p}(1+p^{-s})^{-1}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}={\frac {\zeta (2s)}{\zeta (s)}}.}$

Using their reciprocals, two Euler products for the Möbius function ${\displaystyle \mu (n)}$ are

${\displaystyle \prod _{p}(1-p^{-s})=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}}$

and

${\displaystyle \prod _{p}(1+p^{-s})=\sum _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}={\frac {\zeta (s)}{\zeta (2s)}}.}$

Taking the ratio of these two gives

${\displaystyle \prod _{p}{\Big (}{\frac {1+p^{-s}}{1-p^{-s}}}{\Big )}=\prod _{p}{\Big (}{\frac {p^{s}+1}{p^{s}-1}}{\Big )}={\frac {\zeta (s)^{2}}{\zeta (2s)}}.}$

Since for even s the Riemann zeta function ${\displaystyle \zeta (s)}$ has an analytic expression in terms of a rational multiple of ${\displaystyle \pi ^{s},}$ then for even exponents, this infinite product evaluates to a rational number. For example, since ${\displaystyle \zeta (2)=\pi ^{2}/6,}$ ${\displaystyle \zeta (4)=\pi ^{4}/90,}$ and ${\displaystyle \zeta (8)=\pi ^{8}/9450,}$ then

${\displaystyle \prod _{p}{\Big (}{\frac {p^{2}+1}{p^{2}-1}}{\Big )}={\frac {5}{2}},}$

${\displaystyle \prod _{p}{\Big (}{\frac {p^{4}+1}{p^{4}-1}}{\Big )}={\frac {7}{6}},}$

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

${\displaystyle \prod _{p}(1+2p^{-s}+2p^{-2s}+\cdots )=\sum _{n=1}^{\infty }2^{\omega (n)}n^{-s}={\frac {\zeta (s)^{2}}{\zeta (2s)}},}$

where ${\displaystyle \omega (n)}$ counts the number of distinct prime factors of n, and ${\displaystyle 2^{\omega (n)}}$ is the number of square-free divisors.

If ${\displaystyle \chi (n)}$ is a Dirichlet character of conductor ${\displaystyle N,}$ so that ${\displaystyle \chi }$ is totally multiplicative and ${\displaystyle \chi (n)}$ only depends on n modulo N, and ${\displaystyle \chi (n)=0}$ if n is not coprime to N, then

${\displaystyle \prod _{p}(1-\chi (p)p^{-s})^{-1}=\sum _{n=1}^{\infty }\chi (n)n^{-s}.}$

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

${\displaystyle \prod _{p}(x-p^{-s})\approx {\frac {1}{\operatorname {Li} _{s}(x)}}}$

for ${\displaystyle s>1}$ where ${\displaystyle \operatorname {Li} _{s}(x)}$ is the polylogarithm. For ${\displaystyle x=1}$ the product above is just ${\displaystyle 1/\zeta (s).}$

Many well known constants have Euler product expansions.

The Leibniz formula for π,

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

can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios

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

where each numerator is a prime number and each denominator is the nearest multiple of four.[1]

Other Euler products for known constants include:

Hardy–Littlewood's twin prime constant:

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

Landau-Ramanujan constant:

${\displaystyle {\frac {\pi }{4}}\prod _{p\equiv 1{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{1/2}=0.764223...}$

${\displaystyle {\frac {1}{\sqrt {2}}}\prod _{p\equiv 3{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{-1/2}=0.764223...}$

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

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

Strongly carefree constant ${\displaystyle \times \zeta (2)^{2}}$ OEIS: A065472:

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

Artin's constant OEIS: A005596:

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

Landau's totient constant OEIS: A082695:

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

Carefree constant ${\displaystyle \times \zeta (2)}$ OEIS: A065463:

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

(with reciprocal) OEIS: A065489:

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

Feller-Tornier constant OEIS: A065493:

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

Quadratic class number constant OEIS: A065465:

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

Totient summatory constant OEIS: A065483:

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

Sarnak's constant OEIS: A065476:

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

Carefree constant OEIS: A065464:

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

Strongly carefree constant OEIS: A065473:

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

Stephens' constant OEIS: A065478:

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

Barban's constant OEIS: A175640:

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

Taniguchi's constant OEIS: A175639:

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

Heath-Brown and Moroz constant OEIS: A118228:

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

• 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"

• "Euler product". PlanetMath.
• Hazewinkel, Michiel, ed. (2001) [1994], "Euler product", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Euler Product". MathWorld.
• Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived from the original on 12 Jun 2006.