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{{Short description|Infinite products of functions indexed by primes}}
In [[number theory]], an '''Euler product''' is an expansion of a [[Dirichlet series]] into an [[infinite product]] indexed by [[prime number]]s. The original such product was given for [[Proof of the Euler product formula for the Riemann zeta function|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 {{mvar|a}} is a bounded [[multiplicative function]], then the Dirichlet series

:<math>\sum_{n=1}^\infty \frac{a(n)}{n^s}</math>

is equal to

:<math>\prod_{p\in\mathbb{P}} P(p, s) \quad \text{for } \operatorname{Re}(s) >1 .</math>

where the product is taken over prime numbers {{mvar|p}}, and {{math|''P''(''p'', ''s'')}} is the sum

:<math>\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 </math>

In fact, if we consider these as formal [[generating function]]s, the existence of such a ''formal'' Euler product expansion is a necessary and sufficient condition that {{math|''a''(''n'')}} be multiplicative: this says exactly that {{math|''a''(''n'')}} is the product of the {{math|''a''(''p<sup>k</sup>'')}} whenever {{mvar|n}} factors as the product of the powers {{math|''p<sup>k</sup>''}} of distinct primes {{mvar|p}}.

An important special case is that in which {{math|''a''(''n'')}} is [[totally multiplicative]], so that {{math|''P''(''p'', ''s'')}} is a [[geometric series]]. Then

:<math>P(p, s)=\frac{1}{1-\frac{a(p)}{p^s}},</math>

as is the case for the [[Riemann zeta function]], where {{math|''a''(''n'') {{=}} 1}}, and more generally for [[Dirichlet character]]s.

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

:<math>\operatorname{Re}(s) > C,</math>

that is, in some right [[half-plane]] in the [[complex number]]s. 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 form]]s 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 {{mvar|m}}, and the [[representation theory]] for {{math|GL<sub>''m''</sub>}}.

==Examples==
The following examples will use the notation <math>\mathbb{P}</math> for the set of all primes, that is:
:<math>\mathbb{P}=\{p \in \mathbb{N}\,|\,p\text{ is prime}\}.</math>

The Euler product attached to the [[Riemann zeta function]] {{math|''ζ''(''s'')}}, also using the sum of the geometric series, is

:<math>\begin{align}
\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{align}</math>

while for the [[Liouville function]] {{math|''λ''(''n'') {{=}} (−1)<sup>''ω''(''n'')</sup>}}, it is

:<math> \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)}.</math>

Using their reciprocals, two Euler products for the [[Möbius function]] {{math|''μ''(''n'')}} are

:<math> \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)} </math>

and

:<math> \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)}.</math>

Taking the ratio of these two gives

:<math> \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)}.</math>

Since for even values of {{mvar|s}} the Riemann zeta function {{math|''ζ''(''s'')}} has an analytic expression in terms of a ''rational'' multiple of {{math|''π<sup>s</sup>''}}, then for even exponents, this infinite product evaluates to a rational number. For example, since {{math|''ζ''(2) {{=}} {{sfrac|''π''<sup>2</sup>|6}}}}, {{math|''ζ''(4) {{=}} {{sfrac|''π''<sup>4</sup>|90}}}}, and {{math|''ζ''(8) {{=}} {{sfrac|''π''<sup>8</sup>|9450}}}}, then

:<math>\begin{align}
\prod_{p\, \in\, \mathbb{P}} \left(\frac{p^2+1}{p^2-1}\right) &= \frac53 \cdot \frac{10}{8} \cdot \frac{26}{24} \cdot \frac{50}{48} \cdot \frac{122}{120} \cdots &= \frac{\zeta(2)^2}{\zeta(4)} &= \frac52, \\[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)} &= \frac76, 
\end{align}</math>

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

:<math> \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)}, </math>

where {{math|''ω''(''n'')}} counts the number of distinct prime factors of {{mvar|n}}, and {{math|2<sup>''ω''(''n'')</sup>}} is the number of [[Square-free integer|square-free]] divisors.

If {{math|''χ''(''n'')}} is a Dirichlet character of conductor {{mvar|N}}, so that {{mvar|χ}} is totally multiplicative and {{math|''χ''(''n'')}} only depends on {{math|''n'' mod ''N''}}, and {{math|''χ''(''n'') {{=}} 0}} if {{mvar|n}} is not [[coprime]] to {{mvar|N}}, then

:<math> \prod_{p\, \in\, \mathbb{P}} \frac{1}{1- \frac{\chi(p)}{p^s}} = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.</math>

Here it is convenient to omit the primes {{mvar|p}} dividing the conductor {{mvar|N}} from the product. In his notebooks, Ramanujan generalized the Euler product for the zeta function as

:<math> \prod_{p\, \in\, \mathbb{P}} \left(x-\frac{1}{p^s}\right)\approx \frac{1}{\operatorname{Li}_s (x)} </math>

for {{math|''s'' > 1}} where {{math|Li<sub>''s''</sub>(''x'')}} is the [[polylogarithm]]. For {{math|''x'' {{=}} 1}} the product above is just {{math|{{sfrac|1|''ζ''(''s'')}}}}.

==Notable constants==
Many well known [[constant (mathematics)|constants]] have Euler product expansions.

The [[Leibniz formula for π|Leibniz formula for {{pi}}]]

:<math>\frac{\pi}{4} = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} = 1 - \frac13 + \frac15 - \frac17 + \cdots</math>

can be interpreted as a [[Dirichlet series]] using the (unique) Dirichlet character modulo&nbsp;4, and converted to an Euler product of [[superparticular number|superparticular ratios]] (fractions where numerator and denominator differ by 1):

:<math>\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)=\frac34 \cdot \frac54 \cdot \frac78 \cdot \frac{11}{12} \cdot \frac{13}{12} \cdots,</math>

where each numerator is a prime number and each denominator is the nearest multiple of 4.<ref>{{citation|title=The Legacy of Leonhard Euler: A Tricentennial Tribute|first=Lokenath|last=Debnath|publisher=World Scientific|year=2010|isbn=9781848165267|page=214|url=https://books.google.com/books?id=K2liU-SHl6EC&pg=PA214}}.</ref>

Other Euler products for known constants include:

*The [[Twin prime#First Hardy–Littlewood conjecture|Hardy–Littlewood twin prime constant]]:

::<math> \prod_{p>2} \left(1 - \frac{1}{\left(p-1\right)^2}\right) = 0.660161... </math>

*The [[Landau–Ramanujan constant]]:

::<math>\begin{align}
\frac{\pi}{4} \prod_{p \equiv 1\pmod 4} \left(1 - \frac{1}{p^2}\right)^\frac12 &= 0.764223... \\[6pt]
\frac{1}{\sqrt{2}} \prod_{p \equiv 3\pmod 4} \left(1 - \frac{1}{p^2}\right)^{-\frac12} &= 0.764223... 
\end{align}</math>

* [[Murata's constant]] {{OEIS|A065485}}:

::<math> \prod_{p} \left(1 + \frac{1}{\left(p-1\right)^2}\right) = 2.826419... </math>

* The [[strongly carefree constant]] {{math|×''ζ''(2)<sup>2</sup>}} {{OEIS2C|A065472}}:

::<math> \prod_{p} \left(1 - \frac{1}{\left(p+1\right)^2}\right) = 0.775883... </math>

*[[Artin's conjecture on primitive roots|Artin's constant]] {{OEIS2C|A005596}}:

::<math> \prod_{p} \left(1 - \frac{1}{p(p-1)}\right) = 0.373955... </math>

*[[Landau's totient constant]] {{OEIS2C|A082695}}:

::<math> \prod_{p} \left(1 + \frac{1}{p(p-1)}\right) = \frac{315}{2\pi^4}\zeta(3) = 1.943596... </math>

*The [[carefree constant]] {{math|×''ζ''(2)}} {{OEIS2C|A065463}}:

::<math> \prod_{p} \left(1 - \frac{1}{p(p+1)}\right) = 0.704442... </math>

:and its reciprocal {{OEIS2C|A065489}}:

::<math> \prod_{p} \left(1 + \frac{1}{p^2+p-1}\right) = 1.419562... </math>

*The [[Feller–Tornier constant]] {{OEIS2C|A065493}}:

::<math> \frac{1}{2}+\frac{1}{2} \prod_{p} \left(1 - \frac{2}{p^2}\right) = 0.661317... </math>

*The [[quadratic class number constant]] {{OEIS2C|A065465}}:

::<math> \prod_{p} \left(1 - \frac{1}{p^2(p+1)}\right) = 0.881513... </math>

*The [[totient summatory constant]] {{OEIS2C|A065483}}:

::<math> \prod_{p} \left(1 + \frac{1}{p^2(p-1)}\right) = 1.339784... </math>

*[[Sarnak's constant]] {{OEIS2C|A065476}}:

::<math> \prod_{p>2} \left(1 - \frac{p+2}{p^3}\right) = 0.723648... </math>

*The [[carefree constant]] {{OEIS2C|A065464}}:

::<math> \prod_{p} \left(1 - \frac{2p-1}{p^3}\right) = 0.428249... </math>

*The [[strongly carefree constant]] {{OEIS2C|A065473}}:

::<math> \prod_{p} \left(1 - \frac{3p-2}{p^3}\right) = 0.286747... </math>

*[[Stephens' constant]] {{OEIS2C|A065478}}:

::<math> \prod_{p} \left(1 - \frac{p}{p^3-1}\right) = 0.575959... </math>

*[[Barban's constant]] {{OEIS2C|A175640}}:

::<math> \prod_{p} \left(1 + \frac{3p^2-1}{p(p+1)\left(p^2-1\right)}\right) = 2.596536... </math>

*[[Taniguchi's constant]] {{OEIS2C|A175639}}:

::<math> \prod_{p} \left(1 - \frac{3}{p^3}+\frac{2}{p^4}+\frac{1}{p^5}-\frac{1}{p^6}\right) = 0.678234... </math>

*The [[Heath-Brown–Moroz constant|Heath-Brown and Moroz constant]] {{OEIS2C|A118228}}:

::<math> \prod_{p} \left(1 - \frac{1}{p}\right)^7 \left(1 + \frac{7p+1}{p^2}\right) = 0.0013176... </math>

==Notes==
{{Reflist}}

==References==
{{ref begin}}
* [[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 IANT}} ''(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"
{{ref end}}

==External links==
{{ref begin}}
* {{PlanetMath attribution|urlname=EulerProduct|title=Euler product}}
*{{Eom| title = Euler product | author-last1 =Stepanov| author-first1 = S.A.| oldid = 33842}}
* {{mathworld|urlname=EulerProduct|title=Euler Product}}
* {{Cite web
|last1=Niklasch
|first1=G.
|title=Some number-theoretical constants
|date=23 Aug 2002
|url=http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml
|archiveurl=https://web.archive.org/web/20060612212955/http://gn-50uma.de/alula/essays/Moree/Moree.en.shtml
|archivedate=12 June 2006
}}
{{ref end}}
{{Authority control}}

{{DEFAULTSORT:Euler Product}}
[[Category:Analytic number theory]]
[[Category:Zeta and L-functions]]
[[Category:Mathematical constants]]
[[Category:Infinite products]]