number-theoretic significance

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12: 24.19 Methods of Computation

… ►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. …

13: 27.2 Functions

… ►Other examples of number-theoretic functions treated in this chapter are as follows. … ►

27.2.9

d\left(n\right)=\sum_{d\mathbin{|}n}1

ⓘ

Symbols:: $d_{\NVar{k}}\left(\NVar{n}\right)$ : 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

…

14: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function

f(n)

for large

n

is often difficult to determine because the function values can fluctuate considerably as

n

increases. … ►

27.11.2

\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
A&S Ref:: 24.3.3 III
Referenced by:: §27.11, §27.11, Erratum (V1.1.8) for Section 27.11
Permalink:: http://dlmf.nist.gov/27.11.E2
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

…

15: Bibliography M

… ►

Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

ⓘ

Notes:: A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.
External Links:: Magma
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

S. Makinouchi (1966) Zeros of Bessel functions

J_{\nu}\,(x)

and

Y_{\nu}\,(x)

accurate to twenty-nine significant digits. Technology Reports of the Osaka University 16 (685), pp. 1–44.

ⓘ

Referenced by:: 6th item.
Encodings:: BibTeX

…

16: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number

x

is represented as …where

s

is equal to

1

0

, each

b_{j}

j\geq 1

, is either

0

1

b_{1}

is the most significant bit,

p

(

\in\mathbb{N}

) is the number of significant bits

b_{j}

b_{p-1}

is the least significant bit,

E

is an integer called the exponent,

b_{0}.b_{1}b_{2}\dots b_{p-1}

is the significand, and

f=.b_{1}b_{2}\dots b_{p-1}

is the fractional part. … ►

3.1.2

(-1)^{s}2^{E}\sum_{j=0}^{p-1}b_{j}2^{-j},

ⓘ

Symbols:: $b_{p}$ : bit, $s$ : sign, $p$ : number of significant bits and $E$ : exponent
Permalink:: http://dlmf.nist.gov/3.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §3.1(i), §3.1 and Ch.3

… ►For given values of

E_{{\rm min}}

E_{{\rm max}}

, and

p

, the format width in bits

N

of a computer word is the total number of bits: the sign (one bit), the significant bits

b_{1},b_{2},\dots,b_{p-1}

(

p-1

bits), and the bits allocated to the exponent (the remaining

N-p

bits). … ►

3.1.4

x=(1.b_{1}b_{2}\dots b_{p-1}b_{p}b_{p+1}\dots)\cdot 2^{E},

ⓘ

Symbols:: $b_{p}$ : bit, $p$ : number of significant bits and $E$ : exponent
Permalink:: http://dlmf.nist.gov/3.1.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §3.1(i), §3.1(i), §3.1 and Ch.3

…

… ►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for

\phi

near

\pi/2

with the improvements made in the 1970 reference. …

18: Richard A. Askey

… ►Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …

19: About Color Map

… ►In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of

\pi/2

correponding to the real and imaginary axes, at more immediately recognizable colors. …

20: 19.36 Methods of Computation

… ►If (19.25.9) is used when

0\leq k^{2}\leq 1

, cancellations may lead to loss of significant figures when

k^{2}

is close to 1 and

\phi>\pi/4

, as shown by Reinsch and Raab (2000). … ►

E\left(\phi,k\right)

can be evaluated by using (19.25.7), and

R_{D}

by using (19.21.10), but cancellations may become significant. … ►Near these points there will be loss of significant figures in the computation of

R_{J}

R_{D}

. … ►This method loses significant figures in

\rho

\alpha^{2}

and

k^{2}

are nearly equal unless they are given exact values—as they can be for tables. …