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Euler product

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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 name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.

Contents

Definition

In general, if a is a multiplicative function, then the Dirichlet series

n a ( n ) n s

is equal to

p P ( p , s )

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

1 + a ( p ) p s + a ( p 2 ) p 2 s + .

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 ) = 1 1 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

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 GLm.

Examples

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

p ( 1 p s ) 1 = p ( n = 0 p n s ) = n = 1 1 n s = ζ ( s ) .

while for the Liouville function λ ( n ) = ( 1 ) Ω ( n ) , it is,

p ( 1 + p s ) 1 = n = 1 λ ( n ) n s = ζ ( 2 s ) ζ ( s )

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

p ( 1 p s ) = n = 1 μ ( n ) n s = 1 ζ ( s )

and,

p ( 1 + p s ) = n = 1 | μ ( n ) | n s = ζ ( s ) ζ ( 2 s )

and taking the ratio of these two gives,

p ( 1 + p s 1 p s ) = p ( p s + 1 p s 1 ) = ζ ( s ) 2 ζ ( 2 s )

Since for even 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,

p ( p 2 + 1 p 2 1 ) = 5 2 p ( p 4 + 1 p 4 1 ) = 7 6

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

p ( 1 + 2 p s + 2 p 2 s + ) = n = 1 2 ω ( n ) n s = ζ ( s ) 2 ζ ( 2 s )

where ω ( n ) counts the number of distinct prime factors of n and 2 ω ( n ) 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 modulo N, and χ ( n ) = 0 if n is not coprime to N then,

p ( 1 χ ( p ) p s ) 1 = n = 1 χ ( n ) n s .

Here it is convenient to omit the primes p dividing the conductor N from the product. Ramanujan in his notebooks tried to generalize the Euler product for Zeta function in the form:

p ( x p s ) 1 Li s ( x )

for s > 1 where Li s ( x ) is the polylogarithm. For x = 1 the product above is just 1 / ζ ( s ) .

Notable constants

Many well known constants have Euler product expansions.

The Leibniz formula for π,

π / 4 = n = 0 ( 1 ) n 2 n + 1 = 1 1 3 + 1 5 1 7 + ,

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

π / 4 = ( p 1 ( mod 4 ) p p 1 ) ( p 3 ( mod 4 ) p p + 1 ) = 3 4 5 4 7 8 11 12 13 12 ,

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

Other Euler products for known constants include:

Hardy–Littlewood's twin prime constant:

p > 2 ( 1 1 ( p 1 ) 2 ) = 0.660161...

Landau-Ramanujan constant:

π 4 p = 1 mod 4 ( 1 1 p 2 ) 1 / 2 = 0.764223... 1 2 p = 3 mod 4 ( 1 1 p 2 ) 1 / 2 = 0.764223...

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

p ( 1 + 1 ( p 1 ) 2 ) = 2.826419...

Strongly carefree constant × ζ ( 2 ) 2  A065472:

p ( 1 1 ( p + 1 ) 2 ) = 0.775883...

Artin's constant  A005596:

p ( 1 1 p ( p 1 ) ) = 0.373955...

Landau's totient constant  A082695:

p ( 1 + 1 p ( p 1 ) ) = 315 2 π 4 ζ ( 3 ) = 1.943596...

Carefree constant × ζ ( 2 )  A065463:

p ( 1 1 p ( p + 1 ) ) = 0.704442...

(with reciprocal)  A065489:

p ( 1 + 1 p 2 + p 1 ) = 1.419562...

Feller-Tornier constant  A065493:

1 2 + 1 2 p ( 1 2 p 2 ) = 0.661317...

Quadratic class number constant  A065465:

p ( 1 1 p 2 ( p + 1 ) ) = 0.881513...

Totient summatory constant  A065483:

p ( 1 + 1 p 2 ( p 1 ) ) = 1.339784...

Sarnak's constant  A065476:

p > 2 ( 1 p + 2 p 3 ) = 0.723648...

Carefree constant  A065464:

p ( 1 2 p 1 p 3 ) = 0.428249...

Strongly carefree constant  A065473:

p ( 1 3 p 2 p 3 ) = 0.286747...

Stephens' constant  A065478:

p ( 1 p p 3 1 ) = 0.575959...

Barban's constant  A175640:

p ( 1 + 3 p 2 1 p ( p + 1 ) ( p 2 1 ) ) = 2.596536...

Taniguchi's constant  A175639:

p ( 1 3 p 3 + 2 p 4 + 1 p 5 1 p 6 ) = 0.678234...

Heath-Brown and Moroz constant  A118228:

p ( 1 1 p ) 7 ( 1 + 7 p + 1 p 2 ) = 0.0013176...

References

Euler product Wikipedia


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