Euler product

Template:Short description 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 Template:Mvar 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 Template:Mvar, and Template:Math 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 functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that Template:Math be multiplicative: this says exactly that Template:Math is the product of the Template:Math whenever Template:Mvar factors as the product of the powers Template:Math of distinct primes Template:Mvar.

An important special case is that in which Template:Math is totally multiplicative, so that Template:Math 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 Template:Math, 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

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

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 Template:Mvar, and the representation theory for Template:Math.

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 Template:Math, 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 Template:Math, 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 Template:Math 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 Template:Mvar the Riemann zeta function Template:Math has an analytic expression in terms of a rational multiple of Template:Math, then for even exponents, this infinite product evaluates to a rational number. For example, since Template:Math, Template:Math, and Template:Math, 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 Template:Math counts the number of distinct prime factors of Template:Mvar, and Template:Math is the number of square-free divisors.

If Template:Math is a Dirichlet character of conductor Template:Mvar, so that Template:Mvar is totally multiplicative and Template:Math only depends on Template:Math, and Template:Math if Template:Mvar is not coprime to Template:Mvar, 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 Template:Mvar dividing the conductor Template:Mvar 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 Template:Math where Template:Math is the polylogarithm. For Template:Math the product above is just Template:Math.

Notable constants

Many well known constants have Euler product expansions.

The [[Leibniz formula for π|Leibniz formula for Template: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 4, and converted to an Euler product of 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>Template:Citation.</ref>

Other Euler products for known constants include:

• The 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 (sequence A065485 in the OEIS):

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

• The strongly carefree constant Template:Math Template:OEIS2C:

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

• Artin's constant Template:OEIS2C:

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

• Landau's totient constant Template:OEIS2C:

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

• The carefree constant Template:Math Template:OEIS2C:

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

and its reciprocal Template:OEIS2C:

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

• The Feller–Tornier constant Template:OEIS2C:

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

• The quadratic class number constant Template:OEIS2C:

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

• The totient summatory constant Template:OEIS2C:

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

• Sarnak's constant Template:OEIS2C:

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

• The carefree constant Template:OEIS2C:

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

• The strongly carefree constant Template:OEIS2C:

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

• Stephens' constant Template:OEIS2C:

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

• Barban's constant Template:OEIS2C:

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

• Taniguchi's constant Template:OEIS2C:

<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 and Moroz constant Template:OEIS2C:

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

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)
• Template: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) Template:Isbn (Chapter 17 gives further examples.)
• George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part I, Springer (2005), Template:Isbn
• G. Niklasch, Some number theoretical constants: 1000-digit values"

External links

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