divisor function

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21—26 of 26 matching pages

21: Errata

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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).

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The text “greatest common divisor of $m,n$” was replaced with “greatest common divisor of $k,m$”.

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Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

… ►The Handbook of Mathematical Functions was published, and the Digital Library of Mathematical Functions was released.

22: 3.1 Arithmetics and Error Measures

… ►Division is possible only if the divisor interval does not contain zero. … ►During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length. … ►

3.1.10 $${ϵ}_{\mathrm{𝑟𝑝}}=\left|\ln \left(x^{\ast }/x\right)\right|,$$

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Defines:

${ϵ}_{\mathrm{𝑟𝑝}}$: relative precision (locally)

Symbols:

$\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/3.1.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(v), §3.1 and Ch.3

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23: Bibliography G

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Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Number-Divisor Tables. British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

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External Links:

ZentralBlatt

Referenced by:

§28.26(i), §28.26(i), §28.8(iii).

Encodings:

BibTeX

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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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External Links:

Document

Referenced by:

§19.35(ii).

Encodings:

BibTeX

24: 26.12 Plane Partitions

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§26.12(ii) Generating Functions

… ►where ${\sigma }_2(j)$ is the sum of the squares of the divisors of $j$. … ►

26.12.26 $$\mathrm{pp}\left(n\right)\sim \frac{{\left(\zeta \left(3\right)\right)}^{7/36}}{2^{11/36}{(3\pi )}^{1/2}{n}^{25/36}}\exp \left(3{\left(\zeta \left(3\right)\right)}^{1/3}{\left(\frac{1}{2}n\right)}^{2/3}+{\zeta }^{\prime }\left(-1\right)\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\sim$: asymptotic equality, $\exp z$: exponential function, $\mathrm{pp}\left(n\right)$: number of plane partitions of $n$ , $n$: nonnegative integer and $\pi$: plane partition

Referenced by:

§26.12(iv), Erratum (V1.0.5) for Equation (26.12.26)

Permalink:

http://dlmf.nist.gov/26.12.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

The original statement of this equation, $\mathrm{pp}\left(n\right)\sim {\left(\frac{\zeta \left(3\right)}{2^{11}{n}^{25}}\right)}^{1/36}\exp \left(3{\left(\frac{\zeta \left(3\right)n^2}{4}\right)}^{1/3}+{\zeta }^{\prime }\left(-1\right)\right)$, has been corrected.

Reported 2011-09-05 by Suresh Govindarajan

See also:

Annotations for §26.12(iv), §26.12 and Ch.26

►where $\zeta$ is the Riemann $\zeta$-function (§25.2(i)). …

25: Bibliography W

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Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.

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External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§22.20(ii), §22.20(v).

Encodings:

BibTeX

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

Encodings:

BibTeX

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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26: 23.20 Mathematical Applications

… ► … ►To determine $T$, we make use of the fact that if $(x,y)\in T$ then $y^2$ must be a divisor of $\mathrm{\Delta }$; hence there are only a finite number of possibilities for $y$. Values of $x$ are then found as integer solutions of $x^3+ax+b-y^2=0$ (in particular $x$ must be a divisor of $b-y^2$). … ►

§23.20(v) Modular Functions and Number Theory

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