Hurwitz criterion

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11: 25.21 Software

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§25.21(iv) Hurwitz Zeta Function

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12: 25.15 Dirichlet $L$-functions

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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

Encodings:

pMML, png, TeX

See also:

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

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13: 25.12 Polylogarithms

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25.12.13 $${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}a}\right)+{\mathrm{e}}^{\pi \mathrm{i}s}{\mathrm{Li}}_s\left({\mathrm{e}}^{-2\pi \mathrm{i}a}\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{\pi \mathrm{i}s/2}}{\mathrm{\Gamma }\left(s\right)}\zeta (1-s,a),$$

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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, $(a,b)$: open interval, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Source:

Erdélyi et al. (1953a, (1.11.16), p. 31); with ${\mathrm{Li}}_s\left(z\right)=F(z,s)$, $z={\mathrm{e}}^{2\pi \mathrm{i}a}$

Referenced by:

§25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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14: Bibliography N

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Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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External Links:

ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.15.

Encodings:

BibTeX

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15: Bibliography P

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The Stokes phenomenon associated with the Hurwitz zeta function $\zeta (s,a)$ . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.

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External Links:

ISSN 1364-5021, Document, MathReview (Marc Prevost), ZentralBlatt

Referenced by:

(25.11.43), §25.11(xii).

Encodings:

BibTeX

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16: Bibliography R

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

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17: Bibliography H

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Einige Eigenschaften der Dirichletschen Functionen $F(s)=\sum (\frac{D}{n})\cdot \frac{1}{n}$, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten. Zeitschrift für Math. u. Physik 27, pp. 86–101 (German).

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External Links:

ZentralBlatt

Referenced by:

§25.11(i).

Encodings:

BibTeX

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18: 24.16 Generalizations

… ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

19: Bibliography C

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On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.11(iv), §25.11(iv).

Encodings:

BibTeX

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An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

Encodings:

BibTeX

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20: Software Index

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	Open Source													With Book						Commercial
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25.21(iv) $\zeta (s,a)$	✓				✓	✓						✓	✓	✓			✓				✓	✓	a
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