Euler product
Encyclopedia
In number theory
, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime number
s. The name arose from the case of the Riemann zeta-function, where such a product representation was proved
by Leonhard Euler
.
where a(n) is a multiplicative function
of n may be written as
where the product is taken over prime numbers p and where P(p,s) is the sum
In fact, if we consider these as formal generating function
s, 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(pk) when n factors as the product of the powers pk 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
as is the case for the Riemann zeta-function, where a(n) = 1, and more generally for Dirichlet character
s.
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 form
s 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.
, using also the sum of the geometric series, is
.
while for the Liouville function
, it is,
Using their reciprocals, two Euler products for the Möbius function
are,
and,
and taking the ratio of these two gives,
Since for even s the Riemann zeta function
has an analytic expression in terms of a rational multiple of 
, then for even exponents, this infinite product evaluates to a rational number. For example, since 
, 
, and 
, then,
and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to,
where
counts the number of distinct prime factors of n and 
the number of square-free
divisors.
If
is a Dirichlet character of conductor 
, so that 
is totally multiplicative and 
only depends on n modulo N, and 
if n is not coprime
to N then,
.
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:
for
where 
is the polylogarithm. For 
the product above is just 
have Euler product expansions.
Twin prime constant:
Landau-Ramanujan constant:
Murata's constant :
Strongly carefree constant
:
Artin's constant
:
Landau's totient constant :
Carefree constant
:
(with reciprocal) :
Feller-Tornier constant :
Quadratic class number constant :
Totient summatory constant :
Sarnak's constant :
Carefree constant :
Strongly carefree constant :
Stephens' constant :
Barban's constant :
Taniguchi's constant :
Heath-Brown and Moroz constant :
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. The name arose from the case of the Riemann zeta-function, where such a product representation was proved
Proof of the Euler product formula for the Riemann zeta function
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas' ', published by St Petersburg Academy in 1737.-The Euler product formula:...
by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
.
Definition
In general, a Dirichlet series of the formwhere a(n) is a multiplicative function
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
of n may be written as
where the product is taken over prime numbers p and where P(p,s) is the sum
In fact, if we consider these as formal generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
s, 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(pk) when n factors as the product of the powers pk 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
as is the case for the Riemann zeta-function, where a(n) = 1, and more generally for Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
s.
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 form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s 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
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
for GLm.
Examples
The Euler product attached to the Riemann zeta function, using also the sum of the geometric series, is.while for the Liouville function
, it is,Using their reciprocals, two Euler products for the Möbius function
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
are,and,
and taking the ratio of these two gives,
Since for even s the Riemann zeta function
has an analytic expression in terms of a rational multiple of , then for even exponents, this infinite product evaluates to a rational number. For example, since , , and , then,and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to,
where
counts the number of distinct prime factors of n and the number of square-freeSquare-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
divisors.
If
is a Dirichlet character of conductor , so that is totally multiplicative and only depends on n modulo N, and if n is not coprimeCoprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
to N then,
.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:
for
where is the polylogarithm. For the product above is just Notable constants
Many well known constantsConstant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
have Euler product expansions.
Twin prime constant:
Landau-Ramanujan constant:
Murata's constant :
Strongly carefree constant
:Artin's constant
Artin's conjecture on primitive roots
In number theory, Artin's conjecture on primitive roots states that a given integer a which is not a perfect square and not −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes...
:
Landau's totient constant :
Carefree constant
:(with reciprocal) :
Feller-Tornier constant :
Quadratic class number constant :
Totient summatory constant :
Sarnak's constant :
Carefree constant :
Strongly carefree constant :
Stephens' constant :
Barban's constant :
Taniguchi's constant :
Heath-Brown and Moroz constant :