# §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 and 27

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 and 27

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 $\zeta\left(s\right)$25.2(i)):

 27.4.3 $\zeta\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(1-p^{-s})^{-1},$ $\Re s>1$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: 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 and 27

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 and 27

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 $\displaystyle\sum_{n=1}^{\infty}\mu\left(n\right)n^{-s}$ $\displaystyle=\frac{1}{\zeta\left(s\right)},$ $\Re s>1$, ⓘ Symbols: $\mu\left(\NVar{n}\right)$: Möbius function, $\zeta\left(\NVar{s}\right)$: 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 and 27 27.4.6 $\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)n^{-s}$ $\displaystyle=\frac{\zeta\left(s-1\right)}{\zeta\left(s\right)},$ $\Re s>2$, ⓘ Symbols: $\phi\left(\NVar{n}\right)$: Euler’s totient, $\zeta\left(\NVar{s}\right)$: 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 and 27 27.4.7 $\displaystyle\sum_{n=1}^{\infty}\lambda\left(n\right)n^{-s}$ $\displaystyle=\frac{\zeta\left(2s\right)}{\zeta\left(s\right)},$ $\Re s>1$, ⓘ Symbols: $\lambda\left(\NVar{n}\right)$: Liouville’s function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\Re$: real part and $n$: positive integer Permalink: http://dlmf.nist.gov/27.4.E7 Encodings: TeX, pMML, png See also: Annotations for 27.4 and 27 27.4.8 $\displaystyle\sum_{n=1}^{\infty}|\mu\left(n\right)|n^{-s}$ $\displaystyle=\frac{\zeta\left(s\right)}{\zeta\left(2s\right)},$ $\Re s>1$, ⓘ Symbols: $\mu\left(\NVar{n}\right)$: Möbius function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\Re$: real part and $n$: positive integer Permalink: http://dlmf.nist.gov/27.4.E8 Encodings: TeX, pMML, png See also: Annotations for 27.4 and 27 27.4.9 $\displaystyle\sum_{n=1}^{\infty}2^{\nu\left(n\right)}n^{-s}$ $\displaystyle=\frac{(\zeta\left(s\right))^{2}}{\zeta\left(2s\right)},$ $\Re s>1$, 27.4.10 $\displaystyle\sum_{n=1}^{\infty}d_{k}\left(n\right)n^{-s}$ $\displaystyle=(\zeta\left(s\right))^{k},$ $\Re s>1$, ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $d_{\NVar{k}}\left(\NVar{n}\right)$: divisor function, $\Re$: real part, $k$: positive integer and $n$: positive integer Referenced by: §27.4 Permalink: http://dlmf.nist.gov/27.4.E10 Encodings: TeX, pMML, png See also: Annotations for 27.4 and 27
 27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$, ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$: sum of powers of divisors of $n$, $\Re$: real part and $n$: positive integer A&S Ref: 24.3.3 I.B Permalink: http://dlmf.nist.gov/27.4.E11 Encodings: TeX, pMML, png See also: Annotations for 27.4 and 27
 27.4.12 $\displaystyle\sum_{n=1}^{\infty}\Lambda\left(n\right)n^{-s}$ $\displaystyle=-\frac{\zeta'\left(s\right)}{\zeta\left(s\right)},$ $\Re s>1$, ⓘ Symbols: $\Lambda\left(\NVar{n}\right)$: Mangoldt’s function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\Re$: real part and $n$: positive integer Referenced by: §27.4 Permalink: http://dlmf.nist.gov/27.4.E12 Encodings: TeX, pMML, png See also: Annotations for 27.4 and 27 27.4.13 $\displaystyle\sum_{n=2}^{\infty}(\ln n)n^{-s}$ $\displaystyle=-\zeta'\left(s\right),$ $\Re s>1$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\ln\NVar{z}$: 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 $\ln$ from $\mathrm{log}$. Reported 2015-06-08 See also: Annotations for 27.4 and 27

In (27.4.12) and (27.4.13) $\zeta'\left(s\right)$ is the derivative of $\zeta\left(s\right)$.