Euler product

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 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

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

is equal to

$\prod _{p\in \mathbb{\mathbb{P}}}P(p,s)\text{for }\mathrm{Re}(s)>1.$

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

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

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

$\mathrm{Re}(s)>C,$

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 $\mathbb{\mathbb{P}}$ for the set of all primes, that is:

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

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

$\begin{array}{rl}\prod _{p\in \mathbb{\mathbb{P}}}\left(\frac{1}{1-\frac{1}{p^s}}\right)& =\prod _{p\in \mathbb{\mathbb{P}}}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{1}{p^{ks}}\right)\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}=\zeta (s).\end{array}$

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

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

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

$\prod _{p\in \mathbb{\mathbb{P}}}(1-\frac{1}{p^s})=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n^s}=\frac{1}{\zeta (s)}$

and

$\prod _{p\in \mathbb{\mathbb{P}}}(1+\frac{1}{p^s})=\sum _{n=1}^{\mathrm{\infty }}\frac{|\mu (n)|}{n^s}=\frac{\zeta (s)}{\zeta (2s)}.$

Taking the ratio of these two gives

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

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) = ⁠π²/6⁠, ζ(4) = ⁠π⁴/90⁠, and ζ(8) = ⁠π⁸/9450⁠, then

$\begin{array}{rlrl}\prod _{p\in \mathbb{\mathbb{P}}}\left(\frac{p^2+1}{p^2-1}\right)& =\frac{5}{3}\cdot \frac{10}{8}\cdot \frac{26}{24}\cdot \frac{50}{48}\cdot \frac{122}{120}\cdots & =\frac{\zeta (2{)}^2}{\zeta (4)}& =\frac{5}{2},\\ \prod _{p\in \mathbb{\mathbb{P}}}\left(\frac{p^4+1}{p^4-1}\right)& =\frac{17}{15}\cdot \frac{82}{80}\cdot \frac{626}{624}\cdot \frac{2402}{2400}\cdots & =\frac{\zeta (4{)}^2}{\zeta (8)}& =\frac{7}{6},\end{array}$

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

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

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

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

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

$\frac{\pi }{4}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{2n+1}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$

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):

$\frac{\pi }{4}=\left(\prod _{p\equiv 1(\mathrm{mod}4)}\frac{p}{p-1}\right)\left(\prod _{p\equiv 3(\mathrm{mod}4)}\frac{p}{p+1}\right)=\frac{3}{4}\cdot \frac{5}{4}\cdot \frac{7}{8}\cdot \frac{11}{12}\cdot \frac{13}{12}\cdots ,$

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

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

$\frac{\pi }{2}=\left(\prod _{p\equiv 1(\mathrm{mod}4)}\frac{p}{p+1}\right)\left(\prod _{p\equiv 3(\mathrm{mod}4)}\frac{p}{p-1}\right).$

Taking the ratio of the previous two products gives

$2=\left(\prod _{p\equiv 1(\mathrm{mod}4)}\frac{p-1}{p+1}\right)\left(\prod _{p\equiv 3(\mathrm{mod}4)}\frac{p+1}{p-1}\right).$

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

Other Euler products for known constants include:

• The Hardy–Littlewood twin prime constant:

$\prod _{p>2}(1-\frac{1}{{(p-1)}^2})=0.660161...$

• The Landau–Ramanujan constant:

$\begin{array}{rl}\frac{\pi }{4}\prod _{p\equiv 1(\mathrm{mod}4)}{(1-\frac{1}{p^2})}^{\frac{1}{2}}& =0.764223...\\ \frac{1}{\sqrt{2}}\prod _{p\equiv 3(\mathrm{mod}4)}{(1-\frac{1}{p^2})}^{-\frac{1}{2}}& =0.764223...\end{array}$

• Murata's constant (sequence A065485 in the OEIS):

$\prod _p(1+\frac{1}{{(p-1)}^2})=2.826419...$

• The strongly carefree constant ×ζ(2)² OEIS: A065472:

$\prod _p(1-\frac{1}{{(p+1)}^2})=0.775883...$

• Artin's constant OEIS: A005596:

$\prod _p(1-\frac{1}{p(p-1)})=0.373955...$

• Landau's totient constant OEIS: A082695:

$\prod _p(1+\frac{1}{p(p-1)})=\frac{315}{2{\pi }^4}\zeta (3)=1.943596...$

• The carefree constant ×ζ(2) OEIS: A065463:

$\prod _p(1-\frac{1}{p(p+1)})=0.704442...$

and its reciprocal OEIS: A065489:

$\prod _p(1+\frac{1}{p^2+p-1})=1.419562...$

• The Feller–Tornier constant OEIS: A065493:

$\frac{1}{2}+\frac{1}{2}\prod _p(1-\frac{2}{p^2})=0.661317...$

• The quadratic class number constant OEIS: A065465:

$\prod _p(1-\frac{1}{p^2(p+1)})=0.881513...$

• The totient summatory constant OEIS: A065483:

$\prod _p(1+\frac{1}{p^2(p-1)})=1.339784...$

• Sarnak's constant OEIS: A065476:

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

• The carefree constant OEIS: A065464:

$\prod _p(1-\frac{2p-1}{p^3})=0.428249...$

• The strongly carefree constant OEIS: A065473:

$\prod _p(1-\frac{3p-2}{p^3})=0.286747...$

• Stephens' constant OEIS: A065478:

$\prod _p(1-\frac{p}{p^3-1})=0.575959...$

• Barban's constant OEIS: A175640:

$\prod _p(1+\frac{3p^2-1}{p(p+1)(p^2-1)})=2.596536...$

• Taniguchi's constant OEIS: A175639:

$\prod _p(1-\frac{3}{p^3}+\frac{2}{p^4}+\frac{1}{p^5}-\frac{1}{p^6})=0.678234...$

• The Heath-Brown and Moroz constant OEIS: A118228:

$\prod _p{(1-\frac{1}{p})}^7(1+\frac{7p+1}{p^2})=0.0013176...$

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)
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3  (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

• This article incorporates material from Euler product on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• Hazewinkel, Michiel, ed. (2001), "Euler product", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Weisstein, Eric W.. "Euler Product". http://mathworld.wolfram.com/EulerProduct.html. 
• Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived from the original on 12 June 2006. https://web.archive.org/web/20060612212955/http://gn-50uma.de/alula/essays/Moree/Moree.en.shtml.