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11: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that … ►and $s(h,k)$ is a Dedekind sum given by … ►For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

… ►

12: Karl Dilcher

… ►Dilcher’s research interests include classical analysis, special functions, and elementary, combinatorial, and computational number theory. …

13: Antony Ross Barnett

… ►Barnett’s research interests included number theory and special functions. …

14: Frank Garvan

… ►

27 Functions of Number Theory

15: 27.2 Functions

… ►

§27.2(i) Definitions

… ►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

… ► ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

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Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

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

…

16: 27.10 Periodic Number-Theoretic Functions

… ►is a periodic function of $n(modk)$ and has the finite Fourier-series expansion … ►

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:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. … ► $G(n,\chi )$ is separable for some $n$ if …

17: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

…

18: 27.4 Euler Products and Dirichlet Series

… ►

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. … ►

27.4.4 $$F(s)=\sum _{n=1}^{\mathrm{\infty }}f(n)n^{-s},$$

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Symbols:

$n$: positive integer, $f$: multiplicative function and $F(s)$: generating function

Permalink:

http://dlmf.nist.gov/27.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

…

19: 16.23 Mathematical Applications

… ►

§16.23(iv) Combinatorics and Number Theory

…

20: George E. Andrews

… ►An expert on $q$-series, he is the author of $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. …