number-theoretic functions

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11: 27.2 Functions

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

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

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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

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12: 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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13: 6.16 Mathematical Applications

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►

§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then …where $\pi (x)$ is the number of primes less than or equal to $x$. …

14: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►A similar method can be used for the Euler numbers based on (4.19.5). … ►

§24.19(ii) Values of $B_n$ Modulo $p$

►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 $B_{2n}\equiv 0(modp)$. … ►

Buhler et al. (1992) uses the expansion

24.19.3 $$\frac{t^2}{\cosh t-1}=-2\sum _{n=0}^{\mathrm{\infty }}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$$

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Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $n$: integer and $t$: real or complex

Referenced by:

§24.19(ii)

Permalink:

http://dlmf.nist.gov/24.19.E3

Encodings:

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See also:

Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

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15: Bibliography M

… ►

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

… ►

Methods of Theoretical Physics. Vol. 1, McGraw-Hill Book Co., New York.

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Notes:

Reprinted by Feshbach Publishing, http://www.feshbachpublishing.com/.

External Links:

Feshbach Publishing, MathReview (M. Pinl), ZentralBlatt

Referenced by:

§1.17(iii), §1.5(ii), §12.17, §12.17, §12.17.

Encodings:

BibTeX

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Methods of Theoretical Physics. Vol. 2, McGraw-Hill Book Co., New York.

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Notes:

Reprinted by Feshbach Publishing, http://www.feshbachpublishing.com/.

External Links:

Feshbach Publishing, MathReview (M. Pinl), ZentralBlatt

Referenced by:

§12.17, §12.17.

Encodings:

BibTeX

… ►

Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.9(ii), §30.9(ii).

Encodings:

BibTeX

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Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.9(i), §30.9(i).

Encodings:

BibTeX

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