Dirichlet product (or convolution)

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1: 27.5 Inversion Formulas

… ►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates ►

27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

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Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

Encodings:

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See also:

Annotations for §27.5 and Ch.27

►called the Dirichlet product (or convolution) of $f$ and $g$. …

2: 25.15 Dirichlet $L$-functions

… ► ►

25.15.2 $$L(s,\chi )=\prod _p{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Re }$: real part, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

infinite product, product representation

Source:

Apostol (1976, p. 231)

Permalink:

http://dlmf.nist.gov/25.15.E2

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See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

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See also:

Annotations for §25.15(i), §25.15 and Ch.25

►

25.15.4 $$L(s,\chi )=L(s,{\chi }_0)\prod _{p|k}\left(1-\frac{{\chi }_0(p)}{p^s}\right),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer, $p$: prime number, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

product representation

Source:

Apostol (1976, p. 262)

Referenced by:

§25.15(i)

Permalink:

http://dlmf.nist.gov/25.15.E4

Encodings:

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See also:

Annotations for §25.15(i), §25.15 and Ch.25

…

3: 27.4 Euler Products and Dirichlet Series

§27.4 Euler Products and Dirichlet Series

…

4: 27.8 Dirichlet Characters

… ►Every Dirichlet character $\chi$ (mod $k$) is a product …

5: 27.9 Quadratic Characters

§27.9 Quadratic Characters

… ►The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

6: 27.3 Multiplicative Properties

… ►

27.3.2 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}f(p_r^{a_r}).$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20, §27.3

Permalink:

http://dlmf.nist.gov/27.3.E2

Encodings:

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See also:

Annotations for §27.3 and Ch.27

… ►

27.3.3 $$\varphi \left(n\right)=n\prod _{p|n}(1-p^{-1}),$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

A&S Ref:

24.3.2 I.C

Permalink:

http://dlmf.nist.gov/27.3.E3

Encodings:

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See also:

Annotations for §27.3 and Ch.27

►

27.3.4 $$J_k\left(n\right)=n^k\prod _{p|n}(1-p^{-k}),$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $k$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.3.E4

Encodings:

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See also:

Annotations for §27.3 and Ch.27

►

27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

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Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►Examples are $\lfloor 1/n\rfloor$ and $\lambda \left(n\right)$, and the Dirichlet characters, defined in §27.8. …

7: Bibliography D

… ►

Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

… ►

Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

… ►

Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, pp. 45–81 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band I, pp. 313–342, Reimer, Berlin, 1889 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§25.15(i).

Encodings:

BibTeX

►

Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

Encodings:

BibTeX

… ►

Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

…

8: 25.11 Hurwitz Zeta Function

… ►

25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $k$: nonnegative integer

Referenced by:

§25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.11.E36

Removal (effective with 1.1.4):

This equation has been removed because it is just (25.15.1) combined with (25.15.3).

Clarification (effective with 1.0.23):

The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

See also:

Annotations for §25.11(x), §25.11 and Ch.25

… ►

25.11.37 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}\zeta (nk,a)=-n\ln \mathrm{\Gamma }\left(a\right)+\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(a-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$, $\mathrm{\Re }a\ge 1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

infinite series, series representation

Source:

Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\mathrm{ph}\left(-1\right)=\pi$, which is reasonable since the gamma function is single valued

Permalink:

http://dlmf.nist.gov/25.11.E37

Encodings:

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See also:

Annotations for §25.11(xi), §25.11 and Ch.25

…

9: Errata

… ►

Immediately below (27.11.2), the bound ${\theta }_0$ for Dirichlet’s divisor problem (currently still unsolved) has been changed from $\frac{12}{37}$ Kolesnik (1969) to $\frac{131}{416}$ Huxley (2003).

… ►

25.15.10

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Reported by Gergő Nemes on 2021-08-23

… ►

Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

Reported by Luc Maisonobe on 2021-06-07

… ►

We have emphasized the link with the Dirichlet $L$-function, and used the fact that $\chi (k)=0$. A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).

… ►

Some references were added to §§7.25(ii), 7.25(iii), 7.25(vi), 8.28(ii), and to ¶Products (in §10.74(vii)) and §10.77(ix).

…

10: 25.2 Definition and Expansions

… ►For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. … ►

§25.2(iv) Infinite Products

►

25.2.11 $$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $p$: prime number and $s$: complex variable

Keywords:

infinite product, primes

Source:

Apostol (1976, p. 231)

A&S Ref:

23.2.2

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.2.E11

Encodings:

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See also:

Annotations for §25.2(iv), §25.2 and Ch.25

►product over all primes $p$. …product over zeros $\rho$ of $\zeta$ with $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).