Euler product

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 a is a bounded multiplicative function, then the Dirichlet series

\sum_{n=1}^\infty \frac{a(n)}{n^s}

is equal to

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

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

\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

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

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

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

as is the case for the Riemann zeta function, where $a (n) = 1$ , 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

\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 $GL m$ .

Examples

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

\mathbb{P}=\{p \in \mathbb{N}\,|\,p\text{ is prime}\}.

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

\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 $λ (n) = (-1) Ω(n)$ , it is

\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)}.

Using their reciprocals, two Euler products for the Möbius function $μ (n)$ are

\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)}

and

\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)}.

Taking the ratio of these two gives

\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)}.

Since for even values of s the Riemann zeta function $ζ (s)$ has an analytic expression in terms of a rational multiple of $π s$ , then for even exponents, this infinite product evaluates to a rational number. For example, since $ζ (2) = π 2 ⁄ 6$ , $ζ (4) = π 4 ⁄ 90$ , and $ζ (8) = π 8 ⁄ 9450$ , then

\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

\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)},

where $ω (n)$ counts the number of distinct prime factors of n, and $2 ω (n)$ is the number of square-free divisors.

If $χ (n)$ is a Dirichlet character of conductor N, so that χ is totally multiplicative and $χ (n)$ only depends on $n mod N$ , and $χ (n) = 0$ if n is not coprime to N, then

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

Here it is convenient to omit the primes p dividing the conductor N from the product.

Notable constants

Many well known constants have Euler product expansions.

The Leibniz formula for π

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

\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,

where each numerator is a prime number and each denominator is the nearest multiple of 4.^[1]

Dividing the Euler product for \zeta(2)=\frac{\pi^2}{6} by the previous product, known as Basel problem, one finds that

\frac{\pi}{2} = \left(\prod_{p\equiv 1\pmod 4}\frac{p}{p+1}\right)\left( \prod_{p\equiv 3\pmod 4}\frac{p}{p-1}\right).

Taking the ratio of the previous two products gives

2 = \left(\prod_{p\equiv 1\pmod 4}\frac{p-1}{p+1}\right)\left( \prod_{p\equiv 3\pmod 4}\frac{p+1}{p-1}\right).

The infinite products must be taken in order of increasing primes.

Other Euler products for known constants include:

The Hardy–Littlewood twin prime constant:

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

The Landau–Ramanujan constant:

\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 :

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

The strongly carefree constant $\times ζ (2) 2$ :

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

Artin's constant :

\prod_{p} \left(1 - \frac{1}{p(p-1)}\right) = 0.373955...

Landau's totient constant :

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

The carefree constant $\times ζ (2)$ :

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

and its reciprocal :

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

The Feller–Tornier constant :

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

The quadratic class number constant :

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

The totient summatory constant :

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

Sarnak's constant :

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

The carefree constant :

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

The strongly carefree constant :

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

Stephens' constant :

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

Barban's constant :

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

Taniguchi's constant :

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

The Heath-Brown and Moroz constant :

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

Notes

^ Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.

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)
(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

External links

Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived 12 June 2006 at the Wayback Machine.