The geometric sequences of the reciprocals of the primes equals the harmonic series........................by Euler's product...........with an s of 1...........the number is 2........the next is 3/2.......one and a half........why..................1/1-1/2.........................on the bottom..........1 - 1/2 = 1/2.................1 over 1/2.........is 2........just multiply the top and bottom by two to get rid of the fraction on the bottom............................2 plays and imp. role and so does 1 and a half..............the length of the 5 and 6 star......

Definition[edit]

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.

Convergence[edit]

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 GL_m.

Examples[edit]

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,

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,

.