# §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 $\sum_{n=1}^{\infty}f(n)=\prod_{p}\left(1+\sum_{r=1}^{\infty}f(p^{r})\right),$ Symbols: $n$: positive integer, $p,p_{1},\ldots$: prime numbers and $f$: multiplicative function Permalink: http://dlmf.nist.gov/27.4.E1 Encodings: TeX, pMML, png See also: Annotations for 27.4

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 $\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}.$ Symbols: $n$: positive integer, $p,p_{1},\ldots$: prime numbers and $f$: multiplicative function Permalink: http://dlmf.nist.gov/27.4.E2 Encodings: TeX, pMML, png See also: Annotations for 27.4

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 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(% 1-p^{-s})^{-1},$ $\Re{s}>1$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\Re{}$: real part, $n$: positive integer and $p,p_{1},\ldots$: prime numbers A&S Ref: 23.2.1 and 23.2.2 Permalink: http://dlmf.nist.gov/27.4.E3 Encodings: TeX, pMML, png See also: Annotations for 27.4

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

 27.4.4 $F(s)=\sum_{n=1}^{\infty}f(n)n^{-s},$ Symbols: $n$: positive integer, $f$: multiplicative function and $F(s)$: generating function Permalink: http://dlmf.nist.gov/27.4.E4 Encodings: TeX, pMML, png See also: Annotations for 27.4

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 $\displaystyle\sum_{n=1}^{\infty}\mathop{\mu\/}\nolimits\!\left(n\right)n^{-s}$ $\displaystyle=\frac{1}{\mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\Re{s}>1$, Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: Möbius function, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\Re{}$: real part and $n$: positive integer A&S Ref: 24.3.1 I.B Permalink: http://dlmf.nist.gov/27.4.E5 Encodings: TeX, pMML, png See also: Annotations for 27.4 27.4.6 $\displaystyle\sum_{n=1}^{\infty}\mathop{\phi\/}\nolimits\!\left(n\right)n^{-s}$ $\displaystyle=\frac{\mathop{\zeta\/}\nolimits\!\left(s-1\right)}{\mathop{\zeta% \/}\nolimits\!\left(s\right)},$ $\Re{s}>2$, Symbols: $\mathop{\phi\/}\nolimits\!\left(\NVar{n}\right)$: Euler’s totient, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\Re{}$: real part and $n$: positive integer A&S Ref: 24.3.2 I.B Permalink: http://dlmf.nist.gov/27.4.E6 Encodings: TeX, pMML, png See also: Annotations for 27.4 27.4.7 $\displaystyle\sum_{n=1}^{\infty}\mathop{\lambda\/}\nolimits\!\left(n\right)n^{% -s}$ $\displaystyle=\frac{\mathop{\zeta\/}\nolimits\!\left(2s\right)}{\mathop{\zeta% \/}\nolimits\!\left(s\right)},$ $\Re{s}>1$, 27.4.8 $\displaystyle\sum_{n=1}^{\infty}|\mathop{\mu\/}\nolimits\!\left(n\right)|n^{-s}$ $\displaystyle=\frac{\mathop{\zeta\/}\nolimits\!\left(s\right)}{\mathop{\zeta\/% }\nolimits\!\left(2s\right)},$ $\Re{s}>1$, 27.4.9 $\displaystyle\sum_{n=1}^{\infty}2^{\mathop{\nu\/}\nolimits\!\left(n\right)}n^{% -s}$ $\displaystyle=\frac{(\mathop{\zeta\/}\nolimits\!\left(s\right))^{2}}{\mathop{% \zeta\/}\nolimits\!\left(2s\right)},$ $\Re{s}>1$, 27.4.10 $\displaystyle\sum_{n=1}^{\infty}\mathop{d_{k}\/}\nolimits\!\left(n\right)n^{-s}$ $\displaystyle=(\mathop{\zeta\/}\nolimits\!\left(s\right))^{k},$ $\Re{s}>1$,
 27.4.11 $\sum_{n=1}^{\infty}\mathop{\sigma_{\alpha}\/}\nolimits\!\left(n\right)n^{-s}=% \mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\zeta\/}\nolimits\!\left(s-% \alpha\right),$ $\Re{s}>\max(1,1+\Re{\alpha})$,
 27.4.12 $\displaystyle\sum_{n=1}^{\infty}\mathop{\Lambda\/}\nolimits\!\left(n\right)n^{% -s}$ $\displaystyle=-\frac{\mathop{\zeta\/}\nolimits'\!\left(s\right)}{\mathop{\zeta% \/}\nolimits\!\left(s\right)},$ $\Re{s}>1$, 27.4.13 $\displaystyle\sum_{n=2}^{\infty}(\mathop{\ln\/}\nolimits n)n^{-s}$ $\displaystyle=-\mathop{\zeta\/}\nolimits'\!\left(s\right),$ $\Re{s}>1$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\Re{}$: real part and $n$: positive integer Referenced by: §27.4 Permalink: http://dlmf.nist.gov/27.4.E13 Encodings: TeX, pMML, png Change of Notation (effective with 1.0.10): The notation for logarithm has been changed to $\mathop{\ln\/}\nolimits$ from $\mathrm{log}$. Reported 2015-06-08 See also: Annotations for 27.4

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