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11: 27.2 Functions

… ►

§27.2(i) Definitions

… ►where

p_{1},p_{2},\dots,p_{\nu\left(n\right)}

are the distinct prime factors of

n

, each exponent

a_{r}

is positive, and

\nu\left(n\right)

is the number of distinct primes dividing

n

. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

…

12: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

►

§26.8(i) Definitions

… ► … ►

§26.8(v) Identities

… ►

§26.8(vi) Relations to Bernoulli Numbers

…

13: 34.10 Zeros

… ►In a

\mathit{3j}

symbol, if the three angular momenta

j_{1},j_{2},j_{3}

do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the

\mathit{3j}

symbol is zero. …However, the

\mathit{3j}

and

\mathit{6j}

symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. Such zeros are called nontrivial zeros. ►For further information, including examples of nontrivial zeros and extensions to

\mathit{9j}

symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).

14: 27.13 Functions

… ►

§27.13(i) Introduction

►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

… ►

§27.13(iii) Waring’s Problem

…

15: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime

p

, the Legendre symbol

(n|p)

is defined as follows. … ►

27.9.2

(2|p)=(-1)^{(p^{2}-1)/8}.

ⓘ

Symbols:: $(\NVar{n}|\NVar{p})$ : Legendre symbol and $p$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.9 and Ch.27

►If

p, q

are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer

P

has prime factorization

P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}

, then the Jacobi symbol

(n|P)

is defined by

(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}

, with

(n|1)=1

. …

16: 1.2 Elementary Algebra

… ►The arithmetic mean of

n

numbers

a_{1},a_{2},\dots,a_{n}

is … ►The geometric mean

G

and harmonic mean

H

n

positive numbers

a_{1},a_{2},\dots,a_{n}

are given by … ►If

r

is a nonzero real number, then the weighted mean

M(r)

n

nonnegative numbers

a_{1},a_{2},\dots,a_{n}

, and

n

positive numbers

p_{1},p_{2},\dots,p_{n}

with … ►The dot product notation

\mathbf{u}\cdot\mathbf{v}

is reserved for the physical three-dimensional vectors of (1.6.2). … ►The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. …

17: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

►

§24.2(i) Bernoulli Numbers and Polynomials

… ►

§24.2(ii) Euler Numbers and Polynomials

… ►

Table 24.2.1: Bernoulli and Euler numbers.

► ►►

$n$	$B_{n}$	$E_{n}$
…

► …

18: 27.5 Inversion Formulas

§27.5 Inversion Formulas

… ►Generating functions yield many relations connecting number-theoretic functions. … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: … ►

19: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

… ►

p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. See Table 26.9.1. … ►It follows that

p_{k}\left(n\right)

also equals the number of partitions of

n

into parts that are less than or equal to

k

. …

20: 24.16 Generalizations

§24.16 Generalizations

… ►

Polynomials and Numbers of Integer Order

… ►

Bernoulli Numbers of the Second Kind

… ►

Degenerate Bernoulli Numbers

… ►

§24.16(iii) Other Generalizations

…