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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 adisplaystyle a is a multiplicative function, then the Dirichlet series

∑na(n)n−sdisplaystyle sum _na(n)n^-s,

is equal to

∏pP(p,s)displaystyle prod _pP(p,s),

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

1+a(p)p−s+a(p2)p−2s+⋯.displaystyle 1+a(p)p^-s+a(p^2)p^-2s+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)displaystyle a(n) be multiplicative: this says exactly that a(n)displaystyle a(n) is the product of the a(pk)displaystyle a(p^k) whenever ndisplaystyle n factors as the product of the powers pkdisplaystyle p^k of distinct primes pdisplaystyle p.

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

P(p,s)=11−a(p)p−s,displaystyle P(p,s)=frac 11-a(p)p^-s,

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

Re⁡(s)>C,displaystyle 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 Euler product attached to the Riemann zeta function ζ(s),displaystyle zeta (s), using also the sum of the geometric series, is

∏p(1−p−s)−1=∏p(∑n=0∞p−ns)=∑n=1∞1ns=ζ(s).displaystyle prod _p(1-p^-s)^-1=prod _pBig (sum _n=0^infty p^-nsBig )=sum _n=1^infty frac 1n^s=zeta (s).

while for the Liouville function λ(n)=(−1)Ω(n),displaystyle lambda (n)=(-1)^Omega (n), it is

∏p(1+p−s)−1=∑n=1∞λ(n)ns=ζ(2s)ζ(s).displaystyle prod _p(1+p^-s)^-1=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)displaystyle mu (n) are

∏p(1−p−s)=∑n=1∞μ(n)ns=1ζ(s)displaystyle prod _p(1-p^-s)=sum _n=1^infty frac mu (n)n^s=frac 1zeta (s)

and

∏p(1+p−s)=∑n=1∞|μ(n)|ns=ζ(s)ζ(2s).displaystyle prod _p(1+p^-s)=sum _n=1^infty frac n^s=frac zeta (s)zeta (2s).

Taking the ratio of these two gives

∏p(1+p−s1−p−s)=∏p(ps+1ps−1)=ζ(s)2ζ(2s).displaystyle prod _pBig (frac 1+p^-s1-p^-sBig )=prod _pBig (frac p^s+1p^s-1Big )=frac zeta (s)^2zeta (2s).

Since for even s the Riemann zeta function ζ(s)displaystyle zeta (s) has an analytic expression in terms of a rational multiple of πs,displaystyle pi ^s, then for even exponents, this infinite product evaluates to a rational number. For example, since ζ(2)=π2/6,displaystyle zeta (2)=pi ^2/6, ζ(4)=π4/90,displaystyle zeta (4)=pi ^4/90, and ζ(8)=π8/9450,displaystyle zeta (8)=pi ^8/9450, then

∏p(p2+1p2−1)=52,displaystyle prod _pBig (frac p^2+1p^2-1Big )=frac 52,

∏p(p4+1p4−1)=76,displaystyle prod _pBig (frac p^4+1p^4-1Big )=frac 76,

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

∏p(1+2p−s+2p−2s+⋯)=∑n=1∞2ω(n)n−s=ζ(s)2ζ(2s),displaystyle prod _p(1+2p^-s+2p^-2s+cdots )=sum _n=1^infty 2^omega (n)n^-s=frac zeta (s)^2zeta (2s),

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

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

∏p(1−χ(p)p−s)−1=∑n=1∞χ(n)n−s.displaystyle prod _p(1-chi (p)p^-s)^-1=sum _n=1^infty chi (n)n^-s.

Here it is convenient to omit the primes p dividing the conductor N from the product. In his notebooks, Ramanujan generalized the Euler product for the zeta function as

∏p(x−p−s)≈1Lis⁡(x)displaystyle prod _p(x-p^-s)approx frac 1operatorname Li _s(x)

for s>1displaystyle s>1 where Lis⁡(x)displaystyle operatorname Li _s(x) is the polylogarithm. For x=1displaystyle x=1 the product above is just 1/ζ(s).displaystyle 1/zeta (s).

Notable constants

Many well known constants have Euler product expansions.

The Leibniz formula for π,

π4=∑n=0∞(−1)n2n+1=1−13+15−17+⋯,displaystyle frac pi 4=sum _n=0^infty frac (-1)^n2n+1=1-frac 13+frac 15-frac 17+cdots ,

can be interpreted as a Dirichlet series using the (unique) Dirichlet character modulo 4, and converted to an Euler product of superparticular ratios

π4=(∏p≡1(mod4)pp−1)⋅(∏p≡3(mod4)pp+1)=34⋅54⋅78⋅1112⋅1312⋯,displaystyle frac pi 4=left(prod _pequiv 1pmod 4frac pp-1right)cdot left(prod _pequiv 3pmod 4frac pp+1right)=frac 34cdot frac 54cdot frac 78cdot frac 1112cdot frac 1312cdots ,

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

Other Euler products for known constants include:

Hardy–Littlewood's twin prime constant:

∏p>2(1−1(p−1)2)=0.660161...displaystyle prod _p>2left(1-frac 1(p-1)^2right)=0.660161...

Landau-Ramanujan constant:

π4∏p≡1(mod4)(1−1p2)1/2=0.764223...displaystyle frac pi 4prod _pequiv 1pmod 4left(1-frac 1p^2right)^1/2=0.764223...

12∏p≡3(mod4)(1−1p2)−1/2=0.764223...displaystyle frac 1sqrt 2prod _pequiv 3pmod 4left(1-frac 1p^2right)^-1/2=0.764223...

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

∏p(1+1(p−1)2)=2.826419...displaystyle prod _pleft(1+frac 1(p-1)^2right)=2.826419...

Strongly carefree constant ×ζ(2)2displaystyle times zeta (2)^2 OEIS: A065472:

∏p(1−1(p+1)2)=0.775883...displaystyle prod _pleft(1-frac 1(p+1)^2right)=0.775883...

Artin's constant OEIS: A005596:

∏p(1−1p(p−1))=0.373955...displaystyle prod _pleft(1-frac 1p(p-1)right)=0.373955...

Landau's totient constant OEIS: A082695:

∏p(1+1p(p−1))=3152π4ζ(3)=1.943596...displaystyle prod _pleft(1+frac 1p(p-1)right)=frac 3152pi ^4zeta (3)=1.943596...

Carefree constant ×ζ(2)displaystyle times zeta (2) OEIS: A065463:

∏p(1−1p(p+1))=0.704442...displaystyle prod _pleft(1-frac 1p(p+1)right)=0.704442...

(with reciprocal) OEIS: A065489:

∏p(1+1p2+p−1)=1.419562...displaystyle prod _pleft(1+frac 1p^2+p-1right)=1.419562...

Feller-Tornier constant OEIS: A065493:

12+12∏p(1−2p2)=0.661317...displaystyle frac 12+frac 12prod _pleft(1-frac 2p^2right)=0.661317...

Quadratic class number constant OEIS: A065465:

∏p(1−1p2(p+1))=0.881513...displaystyle prod _pleft(1-frac 1p^2(p+1)right)=0.881513...

Totient summatory constant OEIS: A065483:

∏p(1+1p2(p−1))=1.339784...displaystyle prod _pleft(1+frac 1p^2(p-1)right)=1.339784...

Sarnak's constant OEIS: A065476:

∏p>2(1−p+2p3)=0.723648...displaystyle prod _p>2left(1-frac p+2p^3right)=0.723648...

Carefree constant OEIS: A065464:

∏p(1−2p−1p3)=0.428249...displaystyle prod _pleft(1-frac 2p-1p^3right)=0.428249...

Strongly carefree constant OEIS: A065473:

∏p(1−3p−2p3)=0.286747...displaystyle prod _pleft(1-frac 3p-2p^3right)=0.286747...

Stephens' constant OEIS: A065478:

∏p(1−pp3−1)=0.575959...displaystyle prod _pleft(1-frac pp^3-1right)=0.575959...

Barban's constant OEIS: A175640:

∏p(1+3p2−1p(p+1)(p2−1))=2.596536...displaystyle prod _pleft(1+frac 3p^2-1p(p+1)(p^2-1)right)=2.596536...

Taniguchi's constant OEIS: A175639:

∏p(1−3p3+2p4+1p5−1p6)=0.678234...displaystyle prod _pleft(1-frac 3p^3+frac 2p^4+frac 1p^5-frac 1p^6right)=0.678234...

Heath-Brown and Moroz constant OEIS: A118228:

∏p(1−1p)7(1+7p+1p2)=0.0013176...displaystyle prod _pleft(1-frac 1pright)^7left(1+frac 7p+1p^2right)=0.0013176...

Notes

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, MR 0434929, Zbl 0335.10001
  (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

• 
  "Euler product". PlanetMath.
  


• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Euler product", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• Weisstein, Eric W. "Euler Product". MathWorld.


• 
  Niklasch, G. (23 Aug 2002). "Some number-theoretical constants". Archived from the original on 12 Jun 2006.