# Difference between revisions of "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 $$F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}\ ,\ \ \ s = \sigma + it\ ,\ \ \ \sigma > 1$$ 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.