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11: 27.7 Lambert Series as Generating Functions

§27.7 Lambert Series as Generating Functions

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12: 27.9 Quadratic Characters

§27.9 Quadratic Characters

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13: 24.19 Methods of Computation

… ►For number-theoretic applications it is important to compute $B_{2n}(modp)$ for $2n\le p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0(modp)$. …

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 $$\sum _{n\le x}d\left(n\right)=x\ln x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: 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:

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Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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15: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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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

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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.

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Referenced by:

6th item.

Encodings:

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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16: 20 Theta Functions

Chapter 20 Theta Functions

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17: 19.36 Methods of Computation

… ►If (19.25.9) is used when $0\le k^2\le 1$, cancellations may lead to loss of significant figures when $k^2$ is close to 1 and $\varphi >\pi /4$, as shown by Reinsch and Raab (2000). … ► $E(\varphi ,k)$ 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 $R_J$ or $R_D$. … ►This method loses significant figures in $\rho$ if ${\alpha }^2$ and $k^2$ are nearly equal unless they are given exact values—as they can be for tables. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

18: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number $x$ is represented as …where $s$ is equal to $1$ or $0$, each $b_j$, $j\ge 1$, is either $0$ or $1$, $b_1$ 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, $b_0.b_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the significand, and $f=\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the fractional part. … ►

3.1.2 $${(-1)}^s{2}^E\sum _{j=0}^{p-1}{b}_j{2}^{-j},$$

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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 $E_{\min }$, $E_{\max }$, 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 $b_1,b_2,\mathrm{\dots },b_{p-1}$ ($p-1$ bits), and the bits allocated to the exponent (the remaining $N-p$ bits). … ►

3.1.4 $$x=(1.b_1{b}_2\mathrm{\dots }{b}_{p-1}{b}_p{b}_{p+1}\mathrm{\dots })\cdot 2^E,$$

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Symbols:

$b_p$: bit, $p$: number of significant bits and $E$: exponent

Permalink:

http://dlmf.nist.gov/3.1.E4

Encodings:

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See also:

Annotations for §3.1(i), §3.1(i), §3.1 and Ch.3

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19: 19.38 Approximations

… ►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 $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

20: Bibliography D

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

BibTeX

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Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

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