# §27.4 Euler Products and Dirichlet Series

The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. Every multiplicative $f$ satisfies the identity

 27.4.1 $\sum_{n=1}^{\infty}f(n)=\prod_{p}\left(1+\sum_{r=1}^{\infty}f(p^{r})\right),$

if the series on the left is absolutely convergent. In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes

 27.4.2 $\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}.$

Euler products are used to find series that generate many functions of multiplicative number theory. The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\mathop{\zeta\/}\nolimits\!\left(s\right)$25.2(i)):

 27.4.3 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(% 1-p^{-s})^{-1},$ $\realpart{s}>1$.

The Riemann zeta function is the prototype of series of the form

 27.4.4 $F(s)=\sum_{n=1}^{\infty}f(n)n^{-s},$

called Dirichlet series with coefficients $f(n)$. The function $F(s)$ is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. The following examples have generating functions related to the zeta function:

 27.4.5 $\displaystyle\sum_{n=1}^{\infty}\mathop{\mu\/}\nolimits\!\left(n\right)n^{-s}$ $\displaystyle=\frac{1}{\mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\realpart{s}>1$, 27.4.6 $\displaystyle\sum_{n=1}^{\infty}\mathop{\phi\/}\nolimits\!\left(n\right)n^{-s}$ $\displaystyle=\frac{\mathop{\zeta\/}\nolimits\!\left(s-1\right)}{\mathop{\zeta% \/}\nolimits\!\left(s\right)},$ $\realpart{s}>2$, 27.4.7 $\displaystyle\sum_{n=1}^{\infty}\mathop{\lambda\/}\nolimits\!\left(n\right)n^{% -s}$ $\displaystyle=\frac{\mathop{\zeta\/}\nolimits\!\left(2s\right)}{\mathop{\zeta% \/}\nolimits\!\left(s\right)},$ $\realpart{s}>1$, 27.4.8 $\displaystyle\sum_{n=1}^{\infty}|\mathop{\mu\/}\nolimits\!\left(n\right)|n^{-s}$ $\displaystyle=\frac{\mathop{\zeta\/}\nolimits\!\left(s\right)}{\mathop{\zeta\/% }\nolimits\!\left(2s\right)},$ $\realpart{s}>1$, 27.4.9 $\displaystyle\sum_{n=1}^{\infty}2^{\mathop{\nu\/}\nolimits\!\left(n\right)}n^{% -s}$ $\displaystyle=\frac{(\mathop{\zeta\/}\nolimits\!\left(s\right))^{2}}{\mathop{% \zeta\/}\nolimits\!\left(2s\right)},$ $\realpart{s}>1$, 27.4.10 $\displaystyle\sum_{n=1}^{\infty}\mathop{d_{k}\/}\nolimits\!\left(n\right)n^{-s}$ $\displaystyle=(\mathop{\zeta\/}\nolimits\!\left(s\right))^{k},$ $\realpart{s}>1$,
 27.4.11 $\sum_{n=1}^{\infty}\mathop{\sigma_{\alpha}\/}\nolimits\!\left(n\right)n^{-s}=% \mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\zeta\/}\nolimits\!\left(s-% \alpha\right),$ $\realpart{s}>\max(1,1+\realpart{\alpha})$,
 27.4.12 $\displaystyle\sum_{n=1}^{\infty}\mathop{\Lambda\/}\nolimits\!\left(n\right)n^{% -s}$ $\displaystyle=-\frac{{\mathop{\zeta\/}\nolimits^{\prime}}\!\left(s\right)}{% \mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\realpart{s}>1$, 27.4.13 $\displaystyle\sum_{n=2}^{\infty}(\mathop{\mathrm{log}\,\/}\nolimits n)n^{-s}$ $\displaystyle=-{\mathop{\zeta\/}\nolimits^{\prime}}\!\left(s\right),$ $\realpart{s}>1$.

In (27.4.12) and (27.4.13) ${\mathop{\zeta\/}\nolimits^{\prime}}\!\left(s\right)$ is the derivative of $\mathop{\zeta\/}\nolimits\!\left(s\right)$.