Euler product
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, an Euler product is an expansion of a
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analy ...
into an
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
indexed by
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. The original such product was given for the sum of all positive integers raised to a certain power as proven by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.


Definition

In general, if is a bounded
multiplicative function In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') i ...
, then the Dirichlet series :\sum_ \frac\, is equal to :\prod_ P(p, s) \quad \text \operatorname(s) >1 . where the product is taken over prime numbers , and is the sum :\sum_^\infty \frac = 1 + \frac + \frac + \frac + \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 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 In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
. Then :P(p, s)=\frac, as is the case for the Riemann zeta function, where , and more generally for
Dirichlet character In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1)   \ch ...
s.


Convergence

In practice all the important cases are such that the infinite series and infinite product expansions are
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
in some region :\operatorname(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 , and the
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 studies modules over these abstract algebraic structures. In essen ...
for .


Examples

The following examples will use the notation \mathbb for the set of all primes, that is: :\mathbb=\. The Euler product attached to the Riemann zeta function , also using the sum of the geometric series, is :\begin \prod_ \left(\frac\right) &= \prod_ \left(\sum_^\frac\right) \\ &= \sum_^ \frac = \zeta(s). \end while for the
Liouville function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...
, it is : \prod_ \left(\frac\right) = \sum_^ \frac = \frac. Using their reciprocals, two Euler products for the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
are : \prod_ \left(1-\frac\right) = \sum_^ \frac = \frac and : \prod_ \left(1+\frac\right) = \sum_^ \frac = \frac. Taking the ratio of these two gives : \prod_ \left(\frac\right) = \prod_ \left(\frac\right) = \frac. Since for even values of 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 :\begin \prod_ \left(\frac\right) &= \frac53 \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots &= \frac &= \frac52, \\ pt\prod_ \left(\frac\right) &= \frac \cdot \frac \cdot \frac \cdot \frac \cdots &= \frac &= \frac76, \end and so on, with the first result known by Ramanujan. This family of infinite products is also equivalent to : \prod_ \left(1+\frac+\frac+\cdots\right) = \sum_^\infty \frac = \frac, where counts the number of distinct prime factors of , and is the number of
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
divisors. If is a Dirichlet character of conductor , so that is totally multiplicative and only depends on , and if is not
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to , then : \prod_ \frac = \sum_^\infty \frac. Here it is convenient to omit the primes dividing the conductor from the product. In his notebooks, Ramanujan generalized the Euler product for the zeta function as : \prod_ \left(x-\frac\right)\approx \frac for where is the polylogarithm. For the product above is just .


Notable constants

Many well known constants have Euler product expansions. The Leibniz formula for :\frac = \sum_^\infty \frac = 1 - \frac13 + \frac15 - \frac17 + \cdots can be interpreted as a
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analy ...
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 = \left(\prod_\frac\right)\left( \prod_\frac\right)=\frac34 \cdot \frac54 \cdot \frac78 \cdot \frac \cdot \frac \cdots, where each numerator is a prime number and each denominator is the nearest multiple of 4.. Other Euler products for known constants include: *The Hardy–Littlewood twin prime constant: :: \prod_ \left(1 - \frac\right) = 0.660161... *The
Landau–Ramanujan constant In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number ''b'' that occurs in a theorem proved by Edmund Landau in 1908, stating that for large x, the number of positive integers below x that are t ...
: ::\begin \frac \prod_ \left(1 - \frac\right)^\frac12 &= 0.764223... \\ pt\frac \prod_ \left(1 - \frac\right)^ &= 0.764223... \end * Murata's constant : :: \prod_ \left(1 + \frac\right) = 2.826419... * The strongly carefree constant : :: \prod_ \left(1 - \frac\right) = 0.775883... * Artin's constant : :: \prod_ \left(1 - \frac\right) = 0.373955... * Landau's totient constant : :: \prod_ \left(1 + \frac\right) = \frac\zeta(3) = 1.943596... *The carefree constant : :: \prod_ \left(1 - \frac\right) = 0.704442... :and its reciprocal : :: \prod_ \left(1 + \frac\right) = 1.419562... *The Feller–Tornier constant : :: \frac+\frac \prod_ \left(1 - \frac\right) = 0.661317... *The quadratic class number constant : :: \prod_ \left(1 - \frac\right) = 0.881513... *The totient summatory constant : :: \prod_ \left(1 + \frac\right) = 1.339784... * Sarnak's constant : :: \prod_ \left(1 - \frac\right) = 0.723648... *The carefree constant : :: \prod_ \left(1 - \frac\right) = 0.428249... *The strongly carefree constant : :: \prod_ \left(1 - \frac\right) = 0.286747... * Stephens' constant : :: \prod_ \left(1 - \frac\right) = 0.575959... * Barban's constant : :: \prod_ \left(1 + \frac\right) = 2.596536... * Taniguchi's constant : :: \prod_ \left(1 - \frac+\frac+\frac-\frac\right) = 0.678234... *The Heath-Brown and Moroz constant : :: \prod_ \left(1 - \frac\right)^7 \left(1 + \frac\right) = 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)'' * ''(Provides an introductory discussion of the Euler product in the context of classical number theory.)'' *
G.H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and E.M. Wright, ''An introduction to the theory of numbers'', 5th ed., Oxford (1979) ''(Chapter 17 gives further examples.)'' * George E. Andrews, Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part I'', Springer (2005), * G. Niklasch, ''Some number theoretical constants: 1000-digit values"


External links

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