27 Functions of Number Theory Multiplicative Number Theory 27.3 Multiplicative Properties 27.5 Inversion Formulas

§27.4 Euler Products and Dirichlet Series

ⓘ

Keywords:: Dirichlet series, Euler product, Euler-product representation, Riemann zeta function, generating function, multiplicative number theory, number theory
Notes:: See Apostol (1976, Chapter 11). For (27.4.10) see Titchmarsh (1986b, p. 4).
Referenced by:: §25.2(ii)
Permalink:: http://dlmf.nist.gov/27.4
See also:: Annotations for Ch.27

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 Ch.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 Ch.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 Ch.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 Ch.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 Ch.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 Ch.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 Ch.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 Ch.27
27.4.9	$\displaystyle\sum_{n=1}^{\infty}2^{\nu\left(n\right)}n^{-s}$	$\displaystyle=\frac{(\zeta\left(s\right))^{2}}{\zeta\left(2s\right)},$
$\Re s>1$ ,
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $\Re$ : real part and $n$ : positive integer Permalink: http://dlmf.nist.gov/27.4.E9 Encodings: TeX, pMML, png See also: Annotations for §27.4 and Ch.27
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 Ch.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 a number, $\Re$ : 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 Ch.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 Ch.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 $\mathrm{log}$ . See also: Annotations for §27.4 and Ch.27

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

27.3 Multiplicative Properties 27.5 Inversion Formulas

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§27.4 Euler Products and Dirichlet Series

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