View source for Euler identity

The relation
$$
\sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \left({1 - \frac{1}{p^s} }\right)^{-1}
$$
where $s>1$ is an arbitrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma > 1$.

The Euler identity can be generalized in the form
$$
\sum_{n=1}^\infty f(n) = \prod_p \left({1 - \frac{1}{f(p)} }\right)^{-1}
$$
which holds for every totally-multiplicative arithmetic function $f(n)$> for which the series $\sum_{n=1}^\infty f(n) $ is absolutely convergent.

Another generalization of the Euler identity is the formula
$$
\sum_{n=1}^\infty \frac{a_n}{n^s} = \prod_p \left({1 - a_p p^{-s} + p^{2k-1-2s} }\right)^{-1}
$$
for the [[Dirichlet series|Dirichlet series]]
$$
F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}\ ,\ \ \ s = \sigma + it\ ,\ \ \ \sigma > 1
$$
corresponding to the modular functions
$$
f(z) = \sum_{n=1}^\infty a_n e^{2\pi i n z}
$$
of weight $2k$, which are the eigen functions of the Hecke operator.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Chandrasekharan,   "Introduction to analytic number theory" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,   "Introduction to modular forms" , Springer  (1976)</TD></TR></table>



====Comments====
The product
$$
\prod_p \left({1 - \frac{1}{p^s} }\right)^{-1}
$$
is called the Euler product. For Hecke operators in connection with modular forms see [[Modular form|Modular form]]. For totally-multiplicative arithmetic functions cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of the Riemann zeta-function" , Clarendon Press  (1951)</TD></TR></table>