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31: 27.21 Tables

§27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. … ►Lehmer (1941) gives a comprehensive account of tables in the theory of numbers, including virtually every table published from 1918 to 1941. … ►No sequel to Lehmer (1941) exists to date, but many tables of functions of number theory are included in Unpublished Mathematical Tables (1944).

32: 4.19 Maclaurin Series and Laurent Series

… ►In (4.19.3)–(4.19.9), $B_n$ are the Bernoulli numbers and $E_n$ are the Euler numbers (§§24.2(i)–24.2(ii)). ►

4.19.3 $$\tan z=z+\frac{z^3}{3}+\frac{2}{15}{z}^5+\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.67

Referenced by:

§4.19

Permalink:

http://dlmf.nist.gov/4.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.4 $$\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}{z}^3+\frac{31}{15120}{z}^5+\mathrm{\cdots }+\frac{{(-1)}^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.68

Referenced by:

§4.33

Permalink:

http://dlmf.nist.gov/4.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.5 $$\sec z=1+\frac{z^2}{2}+\frac{5}{24}{z}^4+\frac{61}{720}{z}^6+\mathrm{\cdots }+\frac{{(-1)}^n{E}_{2n}}{(2n)!}{z}^{2n}+\mathrm{\cdots },$$ ,

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sec z$: secant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.69

Referenced by:

§24.1, §24.19(i), §24.2(ii)

Permalink:

http://dlmf.nist.gov/4.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

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4.19.6 $$\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^3}{45}-\frac{2}{945}{z}^5-\mathrm{\cdots }-\frac{{(-1)}^{n-1}{2}^{2n}{B}_{2n}}{(2n)!}{z}^{2n-1}-\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.70

Permalink:

http://dlmf.nist.gov/4.19.E6

Encodings:

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See also:

Annotations for §4.19 and Ch.4

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33: 24.4 Basic Properties

… ►

§24.4(iv) Finite Expansions

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24.4.15 $$B_{2n}=\frac{2n}{2^{2n}(2^{2n}-1)}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{2n-1}{2k}\right)E_{2k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.16 $$E_{2n}=\frac{1}{2n+1}-\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{2n}{2k-1}\right)\frac{2^{2k}(2^{2k-1}-1)B_{2k}}{k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

… ►

§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

34: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

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§24.2(ii) Euler Numbers and Polynomials

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
…

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…

35: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that …where the exponents $1$, $2$, $5$, $7$, $12$, $15$, $\mathrm{\dots }$ are the pentagonal numbers, defined by … ►For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

… ►

36: How to Cite

… ►If you mention a specific equation (or chapter, section, …), you’ll help your readers find it by using our Permalinks & Reference numbers. …produces pdf with the equation and table numbers linking directly into the DLMF website: … ►

Permalinks & Reference numbers

►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …

37: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

38: 27 Functions of Number Theory

Chapter 27 Functions of Number Theory

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39: Karl Dilcher

… ►Dilcher’s research interests include classical analysis, special functions, and elementary, combinatorial, and computational number theory. Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …

40: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►An example is the principal character (mod $k$): … ►If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … ►If $k$ is odd, then the real characters (mod $k$) are the principal character and the quadratic characters described in the next section.