Euler numbers

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21: Software Index

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	Open Source													With Book						Commercial
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24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
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22: 13.27 Mathematical Applications

… ►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. …

23: 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:

pMML, png, TeX

See also:

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

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27.2.7 $$\varphi \left(n\right)={\varphi }_0\left(n\right).$$

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Defines:

$\varphi \left(n\right)$: Euler’s totient

Symbols:

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

Permalink:

http://dlmf.nist.gov/27.2.E7

Encodings:

pMML, png, TeX

See also:

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

►This is the number of positive integers $\le n$ that are relatively prime to $n$; $\varphi \left(n\right)$ is Euler’s totient. … ►

27.2.8 $$a^{\varphi \left(n\right)}\equiv 1(modn),$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\equiv$: modular equivalence, $n$: positive integer and $a$: primitive root

A&S Ref:

24.3.2 II.B

Referenced by:

§27.16, §27.8

Permalink:

http://dlmf.nist.gov/27.2.E8

Encodings:

pMML, png, TeX

See also:

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

… ►The $\varphi \left(n\right)$ numbers $a,a^2,\mathrm{\dots },a^{\varphi \left(n\right)}$ are relatively prime to $n$ and distinct (mod $n$). …

24: 27.4 Euler Products and Dirichlet Series

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27.4.1 $$\sum _{n=1}^{\mathrm{\infty }}f(n)=\prod _p\left(1+\sum _{r=1}^{\mathrm{\infty }}f(p^r)\right),$$

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Symbols:

$n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $f$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

… ►Euler products are used to find series that generate many functions of multiplicative number theory. …

25: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

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26: 25.6 Integer Arguments

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§25.6(i) Function Values

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27: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . …

28: 27.14 Unrestricted Partitions

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27.14.2 $$f\left(x\right)=\prod _{m=1}^{\mathrm{\infty }}(1-x^m)={(x;x)}_{\mathrm{\infty }},$$ ,

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Defines:

$f\left(x\right)$: Euler’s reciprocal function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $m$: positive integer and $x$: real number

Referenced by:

§27.14(iv), §27.21, Erratum (V1.0.28) for Equation (27.14.2)

Permalink:

http://dlmf.nist.gov/27.14.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.28):

The representation in terms of ${(x;x)}_{\mathrm{\infty }}$ was added to this equation.

See also:

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

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27.14.3 $$\frac{1}{f\left(x\right)}=\sum _{n=0}^{\mathrm{\infty }}p\left(n\right)x^n,$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

A&S Ref:

24.2.1 I.B

Permalink:

http://dlmf.nist.gov/27.14.E3

Encodings:

pMML, png, TeX

See also:

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

… ►Euler’s pentagonal number theorem states that ►

27.14.4 $$f\left(x\right)=1-x-x^2+x^5+x^7-x^{12}-x^{15}+\mathrm{\cdots }=1+\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\left(x^{\omega (k)}+x^{\omega (-k)}\right),$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $k$: positive integer, $x$: real number and $\omega (k)$: pentagonal numbers

Permalink:

http://dlmf.nist.gov/27.14.E4

Encodings:

pMML, png, TeX

See also:

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

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27.14.15 $$5\frac{{(f\left(x^5\right))}^5}{{(f\left(x\right))}^6}=\sum _{n=0}^{\mathrm{\infty }}p\left(5n+4\right)x^n$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.14.E15

Encodings:

pMML, png, TeX

See also:

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

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29: 25.16 Mathematical Applications

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25.16.5 $$H\left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{H_n}{n^s},$$

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Symbols:

$H\left(s\right)$: Euler sums, $H_n$: harmonic number, $n$: nonnegative integer, $s$: complex variable and $h(n)$: harmonic number

Keywords:

Euler sum, infinite series, series representation

Source:

Apostol and Vu (1984, p. 85)

Referenced by:

Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.16.E5

Encodings:

pMML, png, TeX

Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_n$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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25.16.10 $$H\left(-2a\right)=\frac{1}{2}\zeta \left(1-2a\right)=-\frac{B_{2a}}{4a},$$ $a=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function and $a$: real or complex parameter

Keywords:

Euler sum, special value

Source:

Apostol and Vu (1984, (7), p. 88)

Permalink:

http://dlmf.nist.gov/25.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. …

30: Bibliography W

… ►

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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