Euler product#Notable constants

In general, if {{mvar|a}} is a bounded multiplicative function, then the Dirichlet series

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{a(n)\}\{n^s\}$

is equal to

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

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

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

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 {{math|a(n)}} be multiplicative: this says exactly that {{math|a(n)}} is the product of the {{math|a(p^k)}} whenever {{mvar|n}} factors as the product of the powers {{math|p^k}} 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

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

as is the case for the Riemann zeta function, where {{math|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

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

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 {{mvar|m}}, and the representation theory for {{math|GL_m}}.

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

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

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

:$\backslash begin\{align\}$

\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}

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

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

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

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

and

:$\backslash prod\_\{p\backslash ,\; \backslash in\backslash ,\; \backslash mathbb\{P\}\}\; \backslash left(1+\backslash frac\{1\}\{p^s\}\backslash right)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac$

Taking the ratio of these two gives

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

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^s}}, then for even exponents, this infinite product evaluates to a rational number. For example, since {{math|ζ(2) {{=}} {{sfrac|π²|6}}}}, {{math|ζ(4) {{=}} {{sfrac|π⁴|90}}}}, and {{math|ζ(8) {{=}} {{sfrac|π⁸|9450}}}}, then

:$\backslash begin\{align\}$

\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}

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

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

where {{math|ω(n)}} counts the number of distinct prime factors of {{mvar|n}}, and {{math|2^ω(n)}} is the number of 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

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

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

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

for {{math|s > 1}} where {{math|Li_s(x)}} is the polylogarithm. For {{math|x {{=}} 1}} the product above is just {{math|{{sfrac|1|ζ(s)}}}}.

Many well known constants have Euler product expansions.

The Leibniz formula for {{pi}}

:$\backslash frac\{\backslash pi\}\{4\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^n\}\{2n+1\}\; =\; 1\; -\; \backslash frac13\; +\; \backslash frac15\; -\; \backslash frac17\; +\; \backslash cdots$

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):

:$\backslash frac\{\backslash pi\}\{4\}\; =\; \backslash left(\backslash prod\_\{p\backslash equiv\; 1\backslash pmod\; 4\}\backslash frac\{p\}\{p-1\}\backslash right)\backslash left(\; \backslash prod\_\{p\backslash equiv\; 3\backslash pmod\; 4\}\backslash frac\{p\}\{p+1\}\backslash right)=\backslash frac34\; \backslash cdot\; \backslash frac54\; \backslash cdot\; \backslash frac78\; \backslash cdot\; \backslash frac\{11\}\{12\}\; \backslash cdot\; \backslash frac\{13\}\{12\}\; \backslash cdots,$

where each numerator is a prime number and each denominator is the nearest multiple of 4.{{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}}.

Other Euler products for known constants include:

• The Hardy–Littlewood twin prime constant:

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

• The Landau–Ramanujan constant:

::$\backslash begin\{align\}$

\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}

• Murata's constant {{OEIS|A065485}}:

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

• The strongly carefree constant {{math|×ζ(2)²}} {{OEIS2C|A065472}}:

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

• Artin's constant {{OEIS2C|A005596}}:

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

• Landau's totient constant {{OEIS2C|A082695}}:

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

• The carefree constant {{math|×ζ(2)}} {{OEIS2C|A065463}}:

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

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

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

• The Feller–Tornier constant {{OEIS2C|A065493}}:

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

• The quadratic class number constant {{OEIS2C|A065465}}:

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

• The totient summatory constant {{OEIS2C|A065483}}:

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

• Sarnak's constant {{OEIS2C|A065476}}:

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

• The carefree constant {{OEIS2C|A065464}}:

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

• The strongly carefree constant {{OEIS2C|A065473}}:

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

• Stephens' constant {{OEIS2C|A065478}}:

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

• Barban's constant {{OEIS2C|A175640}}:

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

• Taniguchi's constant {{OEIS2C|A175639}}:

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

• The Heath-Brown and Moroz constant {{OEIS2C|A118228}}:

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

{{Reflist}}

{{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}}

{{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