§27.4 Euler Products and Dirichlet Series

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

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

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

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 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum _{{n=1}}^{\infty}n^{{-s}}=\prod _{p}(1-p^{{-s}})^{{-1}},$ $\realpart{s}>1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : 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
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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

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 $\sum _{{n=1}}^{\infty}\mathop{\mu\/}\nolimits\!\left(n\right)n^{{-s}}=\frac{1}{\mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\mu\/}\nolimits\!\left(n\right)$ : Möbius function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : 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

27.4.6 $\sum _{{n=1}}^{\infty}\mathop{\phi\/}\nolimits\!\left(n\right)n^{{-s}}=\frac{\mathop{\zeta\/}\nolimits\!\left(s-1\right)}{\mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\realpart{s}>2$ ,

Symbols:: $\mathop{\phi _{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to $n$ , $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : 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

27.4.7 $\sum _{{n=1}}^{\infty}\mathop{\lambda\/}\nolimits\!\left(n\right)n^{{-s}}=\frac{\mathop{\zeta\/}\nolimits\!\left(2s\right)}{\mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\lambda\/}\nolimits\!\left(n\right)$ : Liouville’s function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.4.E7
Encodings:: TeX, pMML, png

27.4.8 $\sum _{{n=1}}^{\infty}|\mathop{\mu\/}\nolimits\!\left(n\right)|n^{{-s}}=\frac{\mathop{\zeta\/}\nolimits\!\left(s\right)}{\mathop{\zeta\/}\nolimits\!\left(2s\right)},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\mu\/}\nolimits\!\left(n\right)$ : Möbius function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.4.E8
Encodings:: TeX, pMML, png

27.4.9 $\sum _{{n=1}}^{\infty}2^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}n^{{-s}}=\frac{(\mathop{\zeta\/}\nolimits\!\left(s\right))^{2}}{\mathop{\zeta\/}\nolimits\!\left(2s\right)},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\nu\/}\nolimits\!\left(n\right)$ : number of distinct primes dividing $n$ , $\realpart{}$ : real part and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.4.E9
Encodings:: TeX, pMML, png

27.4.10 $\sum _{{n=1}}^{\infty}\mathop{d_{{k}}\/}\nolimits\!\left(n\right)n^{{-s}}=(\mathop{\zeta\/}\nolimits\!\left(s\right))^{k},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{d_{{k}}\/}\nolimits\!\left(n\right)$ : divisor function, $\realpart{}$ : 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

27.4.11 $\sum _{{n=1}}^{\infty}\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)n^{{-s}}=\mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\zeta\/}\nolimits\!\left(s-\alpha\right),$ $\realpart{s}>\max(1,1+\realpart{\alpha})$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)$ : sum of powers of divisors of $n$ , $(a,b)$ : open interval, $\realpart{}$ : 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

27.4.12 $\sum _{{n=1}}^{\infty}\mathop{\Lambda\/}\nolimits\!\left(n\right)n^{{-s}}=-\frac{{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s\right)}{\mathop{\zeta\/}\nolimits\!\left(s\right)},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\Lambda\/}\nolimits\!\left(n\right)$ : Mangoldt’s function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part and $n$ : positive integer
Referenced by:: §27.4
Permalink:: http://dlmf.nist.gov/27.4.E12
Encodings:: TeX, pMML, png

27.4.13 $\sum _{{n=2}}^{\infty}(\mathop{\mathrm{log}\,\/}\nolimits n)n^{{-s}}=-{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s\right),$ $\realpart{s}>1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part and $n$ : positive integer
Referenced by:: §27.4
Permalink:: http://dlmf.nist.gov/27.4.E13
Encodings:: TeX, pMML, png

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