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11: 32.9 Other Elementary Solutions

… ►where $P_{n^2+1}(\zeta )$ and $Q_{n^2}(\zeta )$ are polynomials of degrees $n^2+1$ and $n^2$, respectively, with no common zeros. … ►where $P_{n^2-n+1}(\zeta )$ and $Q_{n^2-n}(\zeta )$ are polynomials of degrees $n^2-n+1$ and $n^2-n$, respectively, with no common zeros. …

12: 4.2 Definitions

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4.2.18 $$\ln 10=\mathrm{2.30258\; 50929\; 94045\; 68401}\mathrm{\dots }.$$

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Symbols:

$\ln z$: principal branch of logarithm function

A&S Ref:

4.1.23 (with 10D value.)

Notes:

For more digits see OEIS Sequence A002392; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.2.E18

Encodings:

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See also:

Annotations for §4.2(ii), §4.2 and Ch.4

… ► ${\log }_{10}x$ is the common or Briggs logarithm. …

13: 23.18 Modular Transformations

… ►

23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$.

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\left(m,n\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer

Referenced by:

§23.18, Erratum (V1.0.11) for Equation (23.18.7)

Permalink:

http://dlmf.nist.gov/23.18.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare

See also:

Annotations for §23.18, §23.18 and Ch.23

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14: 26.1 Special Notation

… ► ►►►

$x$	real variable.
…
$\left(h,k\right)$	greatest common divisor of positive integers $h$ and $k$.

…

15: 27.2 Functions

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27.2.6 $${\varphi }_k\left(n\right)=\sum _{\left(m,n\right)=1}{m}^k,$$

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Defines:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number

Symbols:

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

Permalink:

http://dlmf.nist.gov/27.2.E6

Encodings:

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.11 $$J_k\left(n\right)=\sum _{\left(\left(d_1,\mathrm{\dots },d_k\right),n\right)=1}1,$$

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Defines:

$J_k\left(n\right)$: Jordan’s function

Symbols:

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

Referenced by:

§27.2(i)

Permalink:

http://dlmf.nist.gov/27.2.E11

Encodings:

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

►is the number of $k$-tuples of integers $\le n$ whose greatest common divisor is relatively prime to $n$. …

16: 22.8 Addition Theorems

… ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►In the following equations the common modulus $k$ is again suppressed. …

17: 32.8 Rational Solutions

… ►where $P_m(z)$ and $Q_m(z)$ are polynomials of degree $m$, with no common zeros. … ►where $P_{j,n-1}(z)$ and $Q_{j,n}(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. … ►where $\lambda$, $\mu$ are constants, and $P_{n-1}(z)$, $Q_n(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. …

18: 27.14 Unrestricted Partitions

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27.14.10 $$A_k(n)=\sum _{\begin{array}{c}h=1\\ \left(h,k\right)=1\end{array}}^k\exp \left(\pi \mathrm{i}s(h,k)-2\pi \mathrm{i}n\frac{h}{k}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $k$: positive integer, $n$: positive integer, $A_k(n)$: coefficients and $s(h,k)$: Dedekind sum

A&S Ref:

24.2.1 I.C

Permalink:

http://dlmf.nist.gov/27.14.E10

Encodings:

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See also:

Annotations for §27.14(iii), §27.14 and Ch.27

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27.14.19 $$\tau \left(m\right)\tau \left(n\right)=\sum _{d|\left(m,n\right)}{d}^{11}\tau \left(\frac{mn}{d^2}\right),$$ $m,n=1,2,\mathrm{\dots }$.

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Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.14.E19

Encodings:

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See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

19: Mathematical Introduction

… ►

Common Notations and Definitions

…

20: 3.1 Arithmetics and Error Measures

… ►During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length. …