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| In [[number theory]], an '''Euler product''' is an expansion of a [[Dirichlet series]] into an [[infinite product]] indexed by [[prime number]]s. The name arose from the case of the [[Riemann zeta function|Riemann zeta-function]], where such a [[Proof of the Euler product formula for the Riemann zeta function|product representation was proved]] by [[Leonhard Euler]].
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| ==Definition==
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| In general, if <math>a</math> is a [[multiplicative function]], then the Dirichlet series
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| :<math>\sum_{n} a(n)n^{-s}\,</math>
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| is equal to
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| :<math>\prod_{p} P(p, s)\,</math>
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| where the product is taken over prime numbers <math>p</math>, and <math>P(p, s)</math> is the sum
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| :<math>1+a(p)p^{-s} + a(p^2)p^{-2s} + \cdots .</math>
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| 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)</math> be multiplicative: this says exactly that <math>a(n)</math> is the product of the <math>a(p^k)</math> whenever <math>n</math> factors as the product of the powers <math>p^k</math> of distinct primes <math>p</math>.
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| An important special case is that in which <math>a(n)</math> is [[totally multiplicative]], so that <math>P(p, s)</math> is a [[geometric series]]. Then
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| :<math>P(p, s)=\frac{1}{1-a(p)p^{-s}},</math>
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| as is the case for the Riemann zeta-function, where <math>a(n) = 1</math>, and more generally for [[Dirichlet character]]s.
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| ==Convergence==
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| In practice all the important cases are such that the infinite series and infinite product expansions are [[absolutely convergent]] in some region
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| :Re(''s'') > ''C''
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| 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.
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| 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 ''m'', and the [[representation theory]] for GL<sub>''m''</sub>.
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| ==Examples==
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| The Euler product attached to the [[Riemann zeta function]] <math>\zeta(s)</math>, using also the sum of the geometric series, is
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| :<math> \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) </math>.
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| while for the [[Liouville function]] <math>\lambda(n) = (-1)^{\Omega(n)}</math>, it is,
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| :<math> \prod_{p} (1+p^{-s})^{-1} = \sum_{n=1}^{\infty} \frac{\lambda(n)}{n^{s}} = \frac{\zeta(2s)}{\zeta(s)}</math>
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| Using their reciprocals, two Euler products for the [[Möbius function]] <math>\mu(n)</math> are,
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| :<math> \prod_{p} (1-p^{-s}) = \sum_{n=1}^{\infty} \frac{\mu (n)}{n^{s}} = \frac{1}{\zeta(s)} </math>
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| and, | |
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| :<math> \prod_{p} (1+p^{-s}) = \sum_{n=1}^{\infty} \frac{|\mu(n)|}{n^{s}} = \frac{\zeta(s)}{\zeta(2s)}</math>
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| and taking the ratio of these two gives,
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| :<math> \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)} </math>
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| Since for even '''''s''''' the Riemann zeta function <math>\zeta(s)</math> has an analytic expression in terms of a ''rational'' multiple of <math>\pi^{s}</math>, then for even exponents, this infinite product evaluates to a rational number. For example, since <math>\zeta(2)=\pi^2/6</math>, <math>\zeta(4)=\pi^4/90</math>, and <math>\zeta(8)=\pi^8/9450</math>, then,
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| :<math> \prod_{p} \Big(\frac{p^{2}+1}{p^{2}-1}\Big) = \frac{5}{2} </math>
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| :<math> \prod_{p} \Big(\frac{p^{4}+1}{p^{4}-1}\Big) = \frac{7}{6} </math>
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| and so on, with the first result known by [[Ramanujan]]. This family of infinite products is also equivalent to, | |
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| :<math> \prod_{p} (1+2p^{-s}+2p^{-2s}+\cdots) = \sum_{n=1}^{\infty}2^{\omega(n)} n^{-s} = \frac{\zeta(s)^2}{\zeta(2s)} </math>
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| where <math>\omega(n)</math> counts the number of distinct prime factors of ''n'' and <math>2^{\omega(n)}</math> the number of [[Square-free integer|square-free]] divisors.
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| If <math>\chi(n)</math> is a Dirichlet character of ''conductor'' <math>N</math>, so that <math>\chi</math> is totally multiplicative and <math>\chi(n)</math> only depends on ''n'' modulo ''N'', and <math>\chi(n) = 0</math> if ''n'' is not [[coprime]] to ''N'' then,
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| :<math> \prod_{p} (1- \chi(p) p^{-s})^{-1} = \sum_{n=1}^{\infty}\chi(n)n^{-s} </math>.
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| Here it is convenient to omit the primes ''p'' dividing the conductor ''N'' from the product. Ramanujan in his notebooks tried to generalize the Euler product for Zeta function in the form:
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| :<math> \prod_{p} (x-p^{-s})\approx \frac{1}{\operatorname{Li}_{s} (x)} </math>
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| for <math>s > 1</math> where <math>\operatorname{Li}_s(x)</math> is the [[polylogarithm]]. For <math>x=1</math> the product above is just <math> 1/ \zeta (s).</math>
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| ==Notable constants==
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| Many well known [[constant (mathematics)|constants]] have Euler product expansions.
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| The [[Leibniz formula for π]],
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| :<math>\pi/4=\sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1}=1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots,</math>
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| can be interpreted as a [[Dirichlet series]] using the (unique) Dirichlet character modulo 4, and converted to an Euler product of [[superparticular number|superparticular ratios]]
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| :<math>\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,</math> | |
| where each numerator is a prime number and each denominator is the nearest multiple of four.<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=http://books.google.com/books?id=K2liU-SHl6EC&pg=PA214}}.</ref>
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| Other Euler products for known constants include:
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| [[Twin prime constant]]:
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| :<math> \prod_{p>2} \Big(1 - \frac{1}{(p-1)^2}\Big) = 0.660161... </math>
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| [[Landau-Ramanujan constant]]:
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| :<math> \frac{\pi}{4} \prod_{p = 1\,\text{mod}\,4} \Big(1 - \frac{1}{p^2}\Big)^{1/2} = 0.764223... </math>
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| :<math> \frac{1}{\sqrt{2}} \prod_{p = 3\,\text{mod}\,4} \Big(1 - \frac{1}{p^2}\Big)^{-1/2} = 0.764223... </math>
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| [[Murata's constant]] {{OEIS|A065485}}: | |
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| :<math> \prod_{p} \Big(1 + \frac{1}{(p-1)^2}\Big) = 2.826419... </math> | |
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| [[Strongly carefree constant]] <math>\times\zeta(2)^2</math> {{OEIS2C|A065472}}:
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| :<math> \prod_{p} \Big(1 - \frac{1}{(p+1)^2}\Big) = 0.775883... </math>
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| [[Artin's conjecture on primitive roots|Artin's constant]] {{OEIS2C|A005596}}:
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| :<math> \prod_{p} \Big(1 - \frac{1}{p(p-1)}\Big) = 0.373955... </math>
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| [[Landau's totient constant]] {{OEIS2C|A082695}}:
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| :<math> \prod_{p} \Big(1 + \frac{1}{p(p-1)}\Big) = \frac{315}{2\pi^4}\zeta(3) = 1.943596... </math>
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| [[Carefree constant]] <math>\times\zeta(2)</math> {{OEIS2C|A065463}}:
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| :<math> \prod_{p} \Big(1 - \frac{1}{p(p+1)}\Big) = 0.704442... </math>
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| (with reciprocal) {{OEIS2C|A065489}}:
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| :<math> \prod_{p} \Big(1 + \frac{1}{p^2+p-1}\Big) = 1.419562... </math>
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| [[Feller-Tornier constant]] {{OEIS2C|A065493}}:
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| :<math> \frac{1}{2}+\frac{1}{2} \prod_{p} \Big(1 - \frac{2}{p^2}\Big) = 0.661317... </math>
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| [[Quadratic class number constant]] {{OEIS2C|A065465}}:
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| :<math> \prod_{p} \Big(1 - \frac{1}{p^2(p+1)}\Big) = 0.881513... </math>
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| [[Totient summatory constant]] {{OEIS2C|A065483}}:
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| :<math> \prod_{p} \Big(1 + \frac{1}{p^2(p-1)}\Big) = 1.339784... </math>
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| [[Sarnak's constant]] {{OEIS2C|A065476}}:
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| :<math> \prod_{p>2} \Big(1 - \frac{p+2}{p^3}\Big) = 0.723648... </math>
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| [[Carefree constant]] {{OEIS2C|A065464}}:
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| :<math> \prod_{p} \Big(1 - \frac{2p-1}{p^3}\Big) = 0.428249... </math>
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| [[Strongly carefree constant]] {{OEIS2C|A065473}}:
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| :<math> \prod_{p} \Big(1 - \frac{3p-2}{p^3}\Big) = 0.286747... </math>
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| [[Stephens' constant]] {{OEIS2C|A065478}}:
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| :<math> \prod_{p} \Big(1 - \frac{p}{p^3-1}\Big) = 0.575959... </math>
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| [[Barban's constant]] {{OEIS2C|A175640}}:
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| :<math> \prod_{p} \Big(1 + \frac{3p^2-1}{p(p+1)(p^2-1)}\Big) = 2.596536... </math>
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| [[Taniguchi's constant]] {{OEIS2C|A175639}}:
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| :<math> \prod_{p} \Big(1 - \frac{3}{p^3}+\frac{2}{p^4}+\frac{1}{p^5}-\frac{1}{p^6}\Big) = 0.678234... </math>
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| [[Heath-Brown–Moroz constant|Heath-Brown and Moroz constant]] {{OEIS2C|A118228}}:
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| :<math> \prod_{p} \Big(1 - \frac{1}{p}\Big)^7 \Big(1 + \frac{7p+1}{p^2}\Big) = 0.0013176... </math>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [[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)''
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| * {{Apostol IANT}} ''(Provides an introductory discussion of the Euler product in the context of classical number theory.)''
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| * [[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.)''
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| * George E. Andrews, Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part I'', Springer (2005), ISBN 0-387-25529-X
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| * G. Niklasch, ''Some number theoretical constants: 1000-digit values"
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| ==External links==
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| * {{planetmathref|id=5609|title=Euler product}}
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| * {{springer|title=Euler product|id=p/e036560}}
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| * {{mathworld|urlname=EulerProduct|title=Euler Product}}
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| * {{Cite web
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| |last1=Niklasch
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| |first1=G.
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| |title=Some number-theoretical constants
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| |date=23 Aug 2002
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| |url=http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml
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| |archiveurl=http://web.archive.org/web/20060612212955/http://gn-50uma.de/alula/essays/Moree/Moree.en.shtml
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| |archivedate=12 Jun 2006
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| }}
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| {{DEFAULTSORT:Euler Product}}
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| [[Category:Number theory]]
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| [[Category:Zeta and L-functions]]
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| [[Category:Mathematical constants]]
| |
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