relation to Hurwitz zeta function

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1: 25.13 Periodic Zeta Function

… ►Also, …

2: 25.14 Lerch’s Transcendent

… ►The Hurwitz zeta function $\zeta (s,a)$ (§25.11) and the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (§25.12(ii)) are special cases: ►

25.14.2 $$\zeta (s,a)=\mathrm{\Phi }(1,s,a),$$ $\mathrm{\Re }s>1$, $a\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\mathrm{\Re }$: real part, $a$: real or complex parameter and $s$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.11.1), (25.14.1).

Permalink:

http://dlmf.nist.gov/25.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(i), §25.14 and Ch.25

…

3: 25.12 Polylogarithms

… ►and …

4: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: …

5: 25.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,m,n$	nonnegative integers.
…
primes	on function symbols: derivatives with respect to argument.

►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

6: 25.18 Methods of Computation

… ►Calculations relating to derivatives of $\zeta \left(s\right)$ and/or $\zeta (s,a)$ can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). …

7: Bibliography C

… ►

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

Encodings:

BibTeX

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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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8: Bibliography N

… ►

The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-66289-8, MathReview (D. R. Heath-Brown), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

… ►

Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

Encodings:

BibTeX

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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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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

Referenced by:

6th item.

Encodings:

BibTeX

…

9: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

… ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►

A cross index of mathematical software in use at NIST.

10: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

… ► … ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … ►

§25.15(ii) Zeros

… ►Related results are: …