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1: 21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

2: 24.1 Special Notation

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$j,k,\mathrm{\ell },m,n$	integers, nonnegative unless stated otherwise.
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$(k,m)$	greatest common divisor of $k,m$.
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… ►Among various older notations, the most common one is …

3: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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$\left(m,n\right)$	greatest common divisor of $m,n$. If $\left(m,n\right)=1$, $m$ and $n$ are called relatively prime, or coprime.
$\left(d_1,\mathrm{\dots },d_n\right)$	greatest common divisor of $d_1,\mathrm{\dots },d_n$.
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4: 27.3 Multiplicative Properties

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27.3.1 $$f(mn)=f(m)f(n),$$ $\left(m,n\right)=1$.

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Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer, $n$: positive integer and $f(n)$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.3.E1

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Annotations for §27.3 and Ch.27

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27.3.7 $${\sigma }_{\alpha }\left(m\right){\sigma }_{\alpha }\left(n\right)=\sum _{d|\left(m,n\right)}{d}^{\alpha }{\sigma }_{\alpha }\left(\frac{mn}{d^2}\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E7

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Annotations for §27.3 and Ch.27

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27.3.8 $$\varphi \left(m\right)\varphi \left(n\right)=\varphi \left(mn\right)\varphi \left(\left(m,n\right)\right)/\left(m,n\right).$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E8

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Annotations for §27.3 and Ch.27

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5: Viewing DLMF Interactive 3D Graphics

… ►WebGL is supported in the current versions of most common web browsers. …

6: 31.17 Physical Applications

… ►for the common eigenfunction $\mathrm{\Psi }(𝐱)=\mathrm{\Psi }(x_s,x_t,x_u)$, where $a$ is the coupling parameter of interacting spins. … ►

31.17.4 $$\mathrm{\Psi }(𝐱)={(z_1{z}_2)}^{-s-\frac{1}{4}}{((z_1-1)(z_2-1))}^{-t-\frac{1}{4}}{((z_1-a)(z_2-a))}^{-u-\frac{1}{4}}w(z_1)w(z_2),$$

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Symbols:

$z$: complex variable, $a$: complex parameter and $\mathrm{\Psi }(𝐱)$: common eigenfunction

Permalink:

http://dlmf.nist.gov/31.17.E4

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Annotations for §31.17(i), §31.17 and Ch.31

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7: 27.8 Dirichlet Characters

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27.8.4 $$\chi \left(n\right)=0,$$ $\left(n,k\right)>1$.

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E4

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Annotations for §27.8 and Ch.27

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27.8.5

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E5

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Annotations for §27.8 and Ch.27

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27.8.7 $$\chi \left(a\right)=1\text{ for all }a\equiv 1\text{ (mod }d\text{)},$$ $\left(a,k\right)=1$.

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $\equiv$: modular equivalence, $d$: positive integer and $k$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E7

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Annotations for §27.8 and Ch.27

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8: 1.1 Special Notation

… ►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

9: 27.10 Periodic Number-Theoretic Functions

… ►Examples are the Dirichlet characters (mod $k$) and the greatest common divisor $\left(n,k\right)$ regarded as a function of $n$. … ►

27.10.5 $$c_k\left(n\right)=\sum _{d|\left(n,k\right)}d\mu \left(\frac{k}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $c_k\left(n\right)$: Ramanujan’s sum, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.10.E5

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Annotations for §27.10 and Ch.27

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27.10.6 $$s_k(n)=\sum _{d|\left(n,k\right)}f(d)g\left(\frac{k}{d}\right)$$

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Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E6

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Annotations for §27.10 and Ch.27

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27.10.8 $$a_k(m)=\sum _{d|\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac{d}{k}.$$

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Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer, $m$: positive integer, $f(n)$: function, $g(m)$: periodic function and $a_k(m)$: coefficients

Permalink:

http://dlmf.nist.gov/27.10.E8

Encodings:

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Annotations for §27.10 and Ch.27

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10: 10.1 Special Notation

… ►A common alternative notation for $Y_{\nu }\left(z\right)$ is $N_{\nu }(z)$. …