Open main menu

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.

Contents

DefinitionEdit

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

 

is equal to

 

where the product is taken over prime numbers  , and   is the sum

 

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   be multiplicative: this says exactly that   is the product of the   whenever   factors as the product of the powers   of distinct primes  .

An important special case is that in which   is totally multiplicative, so that   is a geometric series. Then

 

as is the case for the Riemann zeta-function, where  , and more generally for Dirichlet characters.

ConvergenceEdit

In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region

 

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.

ExamplesEdit

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   are

 

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   is 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. In his notebooks, Ramanujan generalized the Euler product for the zeta function as

 

for   where   is the polylogarithm. For   the product above is just  

Notable constantsEdit

Many well known constants have Euler product expansions.

The Leibniz formula for π,

 

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

 

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:

 

Landau-Ramanujan constant:

 
 

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

 

Strongly carefree constant   OEISA065472:

 

Artin's constant OEISA005596:

 

Landau's totient constant OEISA082695:

 

Carefree constant   OEISA065463:

 

(with reciprocal) OEISA065489:

 

Feller-Tornier constant OEISA065493:

 

Quadratic class number constant OEISA065465:

 

Totient summatory constant OEISA065483:

 

Sarnak's constant OEISA065476:

 

Carefree constant OEISA065464:

 

Strongly carefree constant OEISA065473:

 

Stephens' constant OEISA065478:

 

Barban's constant OEISA175640:

 

Taniguchi's constant OEISA175639:

 

Heath-Brown and Moroz constant OEISA118228:

 

NotesEdit

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

ReferencesEdit

  • 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 linksEdit