Euler Product - Examples

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 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