divisor%20function

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1: 27.2 Functions

… ►is the number of divisors of $n$ and is the divisor function. It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. …is the number of $k$-tuples of integers $\le n$ whose greatest common divisor is relatively prime to $n$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

2: 26.12 Plane Partitions

… ►

Table 26.12.1: Plane partitions.

► ►►►

$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$
…
3	6	20	75278	37	903 79784
…

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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)). …

3: Bibliography G

… ►

Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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4: 26.10 Integer Partitions: Other Restrictions

… ►

§26.10(ii) Generating Functions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. … ►

§26.10(vi) Bessel-Function Expansion

…

5: Bibliography W

… ►

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 Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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6: Bibliography K

… ►

Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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An improvement of the remainder term in the divisor problem. Mat. Zametki 6, pp. 545–554 (Russian).

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Notes:

In Russian. English translation: Math. Notes 6(1969), no. 5, pp. 784–791.

External Links:

Document, MathReview (A. I. Vinogradov), ZentralBlatt

Referenced by:

§27.11, Erratum (V1.1.8) for Section 27.11.

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

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