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11: 27.18 Methods of Computation: Primes

… ►It runs in time $O\left({(\ln n)}^{c\ln \ln \ln n}\right)$. … ►The AKS (Agrawal–Kayal–Saxena) algorithm is the first deterministic, polynomial-time, primality test. That is to say, it runs in time $O\left({(\ln n)}^c\right)$ for some constant $c$. …

12: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. ►Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time. …

13: 35.4 Partitions and Zonal Polynomials

… ►

35.4.2

ⓘ

Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer, $\kappa$: vector of nonnegative integers and $\mathrm{\ell }(\kappa )$: number of nonzeros

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

… ►

35.4.3 $$Z_{\kappa }\left(𝐓\right)=Z_{\kappa }\left(𝐈\right){\left|𝐓\right|}^{k_m}\underset{𝐎(m)}{\int }\prod _{j=1}^{m-1}{|{(𝐇𝐓{𝐇}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}𝐇,$$ $𝐓\in 𝓢$.

ⓘ

Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

… ►

35.4.5 $$Z_{\kappa }\left(𝐇𝐓{𝐇}^{-1}\right)=Z_{\kappa }\left(𝐓\right),$$ $𝐇\in 𝐎(m)$.

ⓘ

Symbols:

$\in$: element of, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.7 $${\int }_{𝐎(m)}{Z}_{\kappa }\left(𝐒𝐇𝐓{𝐇}^{-1}\right)\mathrm{d}𝐇=\frac{Z_{\kappa }\left(𝐒\right)Z_{\kappa }\left(𝐓\right)}{Z_{\kappa }\left(𝐈\right)}.$$

ⓘ

Symbols:

$\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.8 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{Z}_{\kappa }\left(𝐗\right)d𝐗={\mathrm{\Gamma }}_m\left(a+\kappa \right){\left|𝐓\right|}^{-a}{Z}_{\kappa }\left({𝐓}^{-1}\right),$$

ⓘ

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

…

14: 4.12 Generalized Logarithms and Exponentials

… ►

4.12.7 $$\varphi (x)=\underset{\lfloor x\rfloor \text{ <span class="ltx\_text hilite">times</span>}}{\underbrace{\exp \mathrm{\cdots }\exp }}(x-\lfloor x\rfloor ),$$ $x>1$.

ⓘ

Symbols:

$\exp z$: exponential function, $\lfloor x\rfloor$: floor of $x$ , $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

… ►

4.12.9 $$\psi (x)=\mathrm{\ell }+\underset{\mathrm{\ell }\text{ <span class="ltx\_text hilite">times</span>}}{\underbrace{\ln \mathrm{\cdots }\ln }}x,$$ $x>1$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable, $\psi (x)$: generalized logarithm and $\mathrm{\ell }$: positive integer

Permalink:

http://dlmf.nist.gov/4.12.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

… ►

4.12.10

ⓘ

Defines:

$\mathrm{\ell }$: positive integer (locally)

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.12.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

… ►For $C^{\mathrm{\infty }}$ generalized logarithms, see Walker (1991). …

15: 29.2 Differential Equations

… ►

Figure 29.2.1: $z$-plane: singularities $\times$ $\times$ $\times$ of Lamé’s equation.

…

16: 35.3 Multivariate Gamma and Beta Functions

… ►

35.3.1 $${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

►

35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

ⓘ

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

►

35.3.3 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

ⓘ

Defines:

${\mathrm{B}}_m(a,b)$: multivariate beta function

Symbols:

$\int$: integral, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Keywords:

definition, multivariate beta function

Permalink:

http://dlmf.nist.gov/35.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

… ►

35.3.4 $${\mathrm{\Gamma }}_m\left(a\right)={\pi }^{m(m-1)/4}\prod _{j=1}^m\mathrm{\Gamma }\left(a-\frac{1}{2}(j-1)\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

… ►

35.3.8 $${\mathrm{B}}_m(a,b)={\int }_{𝛀}{\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{-(a+b)}d𝐗,$$ $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

17: 27.16 Cryptography

… ►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers. … ►Procedures for finding such primes require very little computer time. … ►If $p$ and $q$ are known, $s$ and $y^s$ can be determined (mod $n$) by straightforward calculations that require only a few minutes of machine time. …

18: 35.1 Special Notation

… ►All matrices are of order $m\times m$, unless specified otherwise. … ► ►►►►►►

$a,b$	complex variables.
…
$𝟎$	zero matrix.
$𝐈$	identity matrix.
…
$𝐒,𝐓,𝐗$	real symmetric matrices.
…
$𝐇$	orthogonal matrix.
…

…

19: 27.3 Multiplicative Properties

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

►

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

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

►

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

ⓘ

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

… ►

27.3.10 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}{\left(f(p_r)\right)}^{a_r}.$$

ⓘ

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

Permalink:

http://dlmf.nist.gov/27.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

20: 35.6 Confluent Hypergeometric Functions of Matrix Argument

… ►

35.6.2 $$\mathrm{\Psi }(a;b;𝐓)=\frac{1}{{\mathrm{\Gamma }}_m\left(a\right)}{\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{b-a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$, $𝐓\in 𝛀$.

ⓘ

Defines:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind)

Symbols:

$\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

… ►

35.6.4 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b-a\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

►

35.6.5 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{b-\frac{1}{2}(m+1)}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐒𝐗)d𝐗={\mathrm{\Gamma }}_m\left(b\right){\left|𝐈-𝐒{𝐓}^{-1}\right|}^{-a}{\left|𝐓\right|}^{-b},$$ $𝐓>𝐒$, $\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $>$: greater-than for matrices and $m$: positive integer

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

►

35.6.6 $\mathrm{\Re }\left(b_1\right),\mathrm{\Re }\left(b_2\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

… ►

35.6.9 $$\underset{a\to \mathrm{\infty }}{lim}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{\nu +\frac{1}{2}(m+1)};-a^{-1}𝐓)=\frac{A_{\nu }\left(𝐓\right)}{A_{\nu }\left(𝟎\right)}.$$

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(iii), §35.6 and Ch.35

…