completely multiplicative functions

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1: 27.3 Multiplicative Properties

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27.3.8 $$\varphi \left(m\right)\varphi \left(n\right)=\varphi \left(mn\right)\varphi \left(\left(m,n\right)\right)/\left(m,n\right).$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E8

Encodings:

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See also:

Annotations for §27.3 and Ch.27

►A function $f$ is completely multiplicative if $f(1)=1$ and …

2: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►For a completely multiplicative function we use the values at the primes together with (27.3.10). …

3: 27.4 Euler Products and Dirichlet Series

… ►The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): …

4: 27.8 Dirichlet Characters

… ►If $k$ $(>1)$ is a given integer, then a function $\chi \left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. …

5: 22.5 Special Values

… ►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $\mathrm{i}{K}^{\prime }$. …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Such orthonormal sets are called complete. … ► … ►, of unit multiplicity, unless otherwise stated. …and completeness implies … ►and completeness relation …

7: 23.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►►

$𝕃$	lattice in $ℂ$.
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$K\left(k\right)$, $K^{\prime }\left(k\right)$	complete elliptic integrals (§19.2(i)).
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$n\mathbb{Z}$	set of all integer multiples of $n$.
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►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

8: 22.8 Addition Theorems

§22.8 Addition Theorems

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§22.8(iii) Special Relations Between Arguments

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22.8.22 $$z_1+z_2+z_3+z_4=2K\left(k\right).$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E22

Encodings:

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See also:

Annotations for §22.8(iii), §22.8 and Ch.22

… ►If sums/differences of the $z_j$’s are rational multiples of $K\left(k\right)$, then further relations follow. …

9: 27.2 Functions

§27.2 Functions

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§27.2(i) Definitions

… ►They tend to thin out among the large integers, but this thinning out is not completely regular. … ►This is Jordan’s function. … ►This is Liouville’s function. …

10: Bibliography K

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Calculation of the multiple zeros of the derivatives of the cylindrical Bessel functions $J_{\nu }(z)$ and $Y_{\nu }(z)$ . Zh. Vychisl. Mat. i Mat. Fiz. 25 (12), pp. 1749–1760, 1918.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 101–107

External Links:

ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§10.74(vi), 10th item.

Encodings:

BibTeX

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On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:

In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(i), §10.74(vi).

Encodings:

BibTeX

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

Encodings:

BibTeX

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Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).

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Notes:

In Russian. English translation: Soviet Phys. Dokl. 33(3),196–198, (1988).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi).

Encodings:

BibTeX

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Some completely monotonic functions of positive order. Math. Comp. 79 (271), pp. 1697–1707.

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External Links:

ISSN 0025-5718, Document, FullText, MathReview (Necdet Batir), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

Encodings:

BibTeX

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